Cosmic Strings in Hidden Sectors: 1. Radiation of Standard Model Particles
CCosmic Strings in Hidden Sectors: 1. Radiation of Standard Model Particles
Andrew J. Long, ∗ Jeffrey M. Hyde, † and Tanmay Vachaspati ‡ Physics Department, Arizona State University, Tempe, Arizona 85287, USA. (Dated: October 20, 2018)In hidden sector models with an extra
U(1) gauge group, new fields can interact with the StandardModel only through gauge kinetic mixing and the Higgs portal. After the
U(1) is spontaneously bro-ken, these interactions couple the resultant cosmic strings to Standard Model particles. We calculatethe spectrum of radiation emitted by these “dark strings” in the form of Higgs bosons, Z bosons,and Standard Model fermions assuming that string tension is above the TeV scale. We also calculatethe scattering cross sections of Standard Model fermions on dark strings due to the Aharonov-Bohminteraction. These radiation and scattering calculations will be applied in a subsequent paper to studythe cosmological evolution and observational signatures of dark strings.
CONTENTS
I. Introduction 2II. Radiation of Standard Model Particles 5A. Higgs Boson Emission via Linear Coupling 5B. Higgs Boson Emission via Quadratic Coupling 9C. Z-Boson Emission via Linear Coupling 12D. Z Boson Emission via Quadratic Coupling 14E. Fermion Emission via Aharonov-Bohm Coupling 16III. Scattering Cross Sections 20IV. Summary and Conclusion 21Acknowledgments 25A. Worldsheet Formalism 25B. Calculation of Particle Radiation from the String 261. Scalar Radiation via Linear Coupling 272. Scalar Radiation via Quadratic Coupling 283. Vector Radiation via Linear Coupling 284. Vector Radiation via Quadratic Coupling 305. Dirac Spinor Radiation – Direct Coupling 30 ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - ph ] M a y
6. Dirac Spinor Radiation – AB Coupling 31C. Calculation of the Worldsheet Integrals 321. Saddle Point Integral 332. Discontinuity Integral 36D. Scalar and Tensor Integrals for Cusps, Kinks, and Kink Collisions 371. Scalar Integral – Cusp 382. Scalar Integral – Kink 393. Scalar Integral – Kink Collision 404. Tensor Integral – Cusp 405. Tensor Integral – Kink 416. Tensor Integral – Kink Collision 41E. Cusp Boost Factor and UV Sensitivity 42References 43
I. INTRODUCTION
In the Standard Model of particle physics, all of the matter fields are charged under the gauge groupof the theory, and consequently all of the particles participate in the gauge interactions. It is natural to askwhether there can be new particles that do not participate in any of the Standard Model gauge interactions,and whose fields are singlets under the Standard Model gauge group. Such fields would be sequesteredin a “hidden sector” where they participate in their own gauge interactions under which the SM fields aresinglets. Despite their minimal nature, hidden sector models admit a rich phenomenology; they have beenwell-studied in the context of collider physics [1–5] as well as dark matter [6–11]. In Refs. [12, 13] wehave pointed out that these models may also contain cosmic string solutions, called “dark strings”, that havenovel interactions with Standard Model fields. The aim of the present paper is to derive the radiative andscattering properties of these strings. In a subsequent paper we will use these properties to study potentialastrophysical and cosmological signatures of dark strings.The Lagrangian for the hidden sector model under consideration is of the form L = L SM + L HS + L int . (I.1)The first term, L SM , is the Standard Model (SM) Lagrangian; the second term, L HS = | D µ S | −
14 ˆ X µν − κ ( S ∗ S − σ ) , (I.2)is the hidden sector (HS) Lagrangian with S a complex scalar field charged under a U(1) X gauge groupthat has ˆ X µ as its gauge potential, D µ = ∂ µ − ig X ˆ X µ ; and the third term, L int = − α (Φ † Φ − η )( S ∗ S − σ ) − sin (cid:15) X µν Y µν , (I.3)is the interaction Lagrangian with Φ the Higgs doublet and Y µ the hypercharge gauge field. The massscale of the hidden sector fields is set by the parameter σ , and η = 174 GeV is the vacuum expectationvalue (VEV) of the Higgs field. The two terms in L int are called the Higgs portal (HP) term [14] and thegauge-kinetic mixing (GKM) term [15, 16], respectively. For σ (cid:46) TeV , the HP and GKM couplings arewell-constrained, | α | , | sin (cid:15) | (cid:28) [17, 18], but if σ is above the TeV scale, making HS particles inaccessibleat laboratory energies, the hidden sector model is (as yet) unconstrained. In principle the hidden sector canbe extended to include additional fields and interactions; we retain only the minimal degrees of freedomnecessary to study radiation of SM particles from the cosmic string.The VEV (cid:104) S (cid:105) = σ spontaneously breaks the U(1) X completely. Consequently the model admitstopological (cosmic) string solutions [19]. The string tension is set by the symmetry breaking mass scale µ ≈ σ , and we will use M ≡ √ µ ≈ σ through the text. In Ref. [13] we studied these “dark string”solutions, which were found to contain a non-trivial structure in the dark sector fields, S and ˆ X µ , as well asin the SM fields, Φ and Y µ . (See also [20] for the case when sin (cid:15) = 0 .) In the decoupling limit, σ (cid:29) η , thedark fields form a thin core of thickness on the order of σ − , and the SM fields form a wide dressing withthickness η − . The dressing arises because the string core sources the SM Higgs and Z boson fields, φ H and Z µ . In the limit σ (cid:29) η we can integrate out the heavy HS fields leaving only the zero thickness stringcore. In Ref. [13] we found the effective interaction of the string core with the light SM fields to be S (1)int = g H str η (cid:90) d σ √− γ φ H ( X µ ) + g Z str (cid:16) ησ (cid:17) (cid:90) dσ µν Z µν ( X µ ) (I.4)where X µ denotes the location of the zero thickness string core, and the rest of the notation is defined inAppendix A. The coupling constants g H str and g Z str have been derived in Ref. [13] in terms of α , sin (cid:15) , andother Lagrangian coupling constants. We shall treat them as free parameters in the present paper. Note thatthe interaction in Eq. (I.4) is valid for σ (cid:29) η , when the string core is much thinner than the SM dressing. Ifthe core and dressing widths are comparable, the effective interaction formalism breaks down and the fullfield theory equations must be solved to evaluate string-particle interactions.The linear interactions given above arise because the Higgs gets a VEV, and the string acts asa source that modifies the VEV. In addition, the string also couples to the SM fields through the moregeneric quadratic interactions. Upon integrating out the heavy hidden sector fields, the effective quadratic Cusp Kink Kink (cid:45)
Kink Collision
FIG. 1: Illustrations of the cusp, kink, and kink-kink collision. A cusp is a point on the loop thatinstantaneously moves at the speed of light; a kink is a discontinuity in the tangent vector to the string thatmoves around the loop in one direction; a kink-kink collision occurs when two oppositely moving kinkscollide.interactions for the Higgs and Z boson are S (2)int = g HH str (cid:90) d σ √− γ φ H ( X ) + g ZZ str (cid:16) ησ (cid:17) (cid:90) d σ √− γ Z µ ( X ) Z µ ( X ) . (I.5)The quadratic Higgs interaction derives directly from the HP term in Eq. (I.3), and we can estimate g HH str ≈ α up to order one factors related to integrals of the profile functions. The quadratic Z boson interaction resultsfrom the mixing of the Z boson with the heavy ˆ X µ field. The mixing angle goes like (sin (cid:15) )( η/σ ) [13],and therefore we obtain the quadratic interaction in Eq. (I.5) with g ZZ str ≈ sin (cid:15) . The W bosons will have acoupling similar to the Z boson coupling in Eq. (I.5), and our results for the Z bosons carry over to the otherweak bosons as well. The remaining bosonic SM fields, the gluons and the photons, do not couple to thestring worldsheet at leading order [13].In addition to interactions with φ H and Z µ , the string also couples to the SM fermions due to anAharonov-Bohm (AB) interaction [21]. Upon circumnavigating the string on a length scale larger than thewidth of the SM dressing fields, the fermion wavefunction picks up a phase that is π times [13] θ q = − θ W sin (cid:15)g X q. (I.6)where q is the electromagnetic charge on the fermion, and θ W is the weak mixing angle. The AB interactionis topological, insensitive to the details of the structure of the string, and in particular, does not assume σ (cid:29) η . By virtue of the interactions in Eqs. (I.4) and (I.5), dynamical dark strings will emit Higgs and Zbosons, and it will emit SM fermions through the AB interaction. In the following sections, we calculatethe spectrum of radiation in the form of Higgs and Z bosons that is emitted from cusps, kinks, and kinkcollisions on cosmic string loops (see Fig. 1). The scalar boson radiation channels have been derived pre-viously [22–25]. We refine these calculations by carefully estimating all dimensionless coefficients, and insome cases also correcting errors. Most importantly, we find that the calculation of Ref. [22] underestimatesthe scalar radiation by a factor of √ M L (cid:29) , which arises because the radiation from the cusp is highlyboosted. The vector boson channels have not been worked out previously, and we present them here for thefirst time. We also estimate radiation from the Aharonov-Bohm interaction by drawing on results from theliterature. Our results, it should be emphasized, are not unique to the dark string model; instead, the spectraderived here apply to any model with effective interactions of the form in Eqs. (I.4) and (I.5).Particle radiation is expected to play an important role in the evolution of light cosmic string forwhich gravitational radiation is suppressed. Specifically, we find that Higgs boson emission is the dominantenergy loss mechanism for light dark strings. The emission of SM particles may also lead to observationalsignatures of dark strings through astrophysics or cosmology, and we will explore this possibility in acompanion paper [26]. II. RADIATION OF STANDARD MODEL PARTICLES
The interactions in Eqs. (I.4) and (I.5) allow a dark string to emit Higgs and Z bosons, and SMfermions are radiated by virtue of the non-local Aharanov-Bohm interaction. In the subsections below wefirst present the spectrum of Higgs and Z boson radiation from a general string configuration, and we thenspecify to the cases of cusps, kinks, and kink-kink collisions as these are expected to the be the three mostcopious sources of particle radiation. We leave the details of these calculations to the Appendices.
A. Higgs Boson Emission via Linear Coupling
The physical Higgs field, φ H ( x ) , couples to the dark string through the effective interaction S H int = g H str η (cid:90) d σ √− γ φ H ( X ) . (II.1)Since this term is linear in φ H it acts as a classical source term for the Higgs field and leads to radiationfrom the string. Note that the dimensional prefactor, η ≈
174 GeV , is the vacuum expectation value of theHiggs field; this interaction would not be present if not for electroweak symmetry breaking. In Appendix Bwe calculate the spectrum of Higgs boson radiation for a string loop. Taking A = g H str η in Eq. (B.7) we find dN H = ( g H str η ) |I ( k ) | | k | dωd Ω2(2 π ) (II.2)where the integral I ( k ) = (cid:90) d σ √− γ e ik · X , (II.3)is a functional of the string worldsheet, X µ ( τ, σ ) , that describes the motion of the string loop. The kine-matical variables are defined by k µ = (cid:8) ω , k (cid:9) with ω = ( m H + | k | ) / and m H the Higgs boson mass.In the following subsections we specify X µ so as to evaluate the spectrum and total power of Higgs bosonemission from cusps, kinks, and kink-kink collisions.
1. Higgs Emission from a Cusp
A cusp occurs when there is a point on the worldsheet where ∂ σ X = 0 . At this point the velocityof the string segment approaches the speed of light, and the radiation is highly boosted. In the rest frame ofthe loop, the momentum of the emitted radiation cannot exceed the inverse string thickness, i.e. | k | < M where M = √ µ , else the point-like interaction in Eq. (II.1) is inapplicable, and the radiation is suppressed.However, due to the large boost factor, γ boost ∼ √ M L , the radiation does not cut off until | k | ≈ M √ M L
Appendix E.Inserting the scalar integral from Eq. (D.12) into the spectrum in Eq. (II.2) we find dN (cusp) H = ( g H str η ) π ) S (cusp) L / | k | / dω d Ω ,ψ m H (cid:112) m H L < | k | < M √ M L , θ <
Θ ( | k | L ) − / (cone) . (II.4)where ψ ≈ . (see Eq. (C.18)), Θ ≈ . (below Eq. (C.13)), and . (cid:46) S (cusp) (cid:46) (see belowEq. (D.12)). As explained above, the spectrum is cutoff in the UV by the (boosted) string thickness, andit cuts off in the IR due to a destructive interference that is manifest in the breakdown of the saddle pointapproximation. Since typically m H L (cid:29) , the radiation is ultra-relativistic and we can approximate | k | ≈ ω and d | k | ≈ dω .The radiation is emitted into a cone that has an opening angle Θ( | k | L ) − / . Integrating over thesolid angle, we find the spectrum to be dN (cusp)H ≈ ( g H str η ) π ) Θ S (cusp) L / d | k || k | / , ψ m H (cid:112) m H L < | k | < M √ M L . (II.5)The total energy emitted from a cusp is E (cusp) H = (cid:90) ω dN (cusp) H = 3( g H str η ) π ) ψ − / Θ S (cusp) (cid:114) Lm H (cid:18) − ψ / (cid:114) m H M (cid:19) . (II.6)Since we are interested in heavy strings, M (cid:29) m H , we can neglect the second term in the parenthesis. Ifcusps appear on a loop with frequency f c /T where T = L/ is the loop oscillation period, then the averagepower emitted per oscillation is P H = 2 E H f c /L , or P (cusp) H = Γ (cusp) H ( g H str η ) √ m H L (II.7)where Γ (cusp) H ≡ π ) ψ − / Θ f c S (cusp) . Assuming f c ≈ , the dimensionless coefficient takes values inthe range − (cid:46) Γ (cusp) H (cid:46) − . This result agrees with a previous calculation in the literature [24].
2. Higgs Emission from a Kink
A kink occurs where there is a discontinuity in the derivative of the string worldsheet ∂ σ X . Weobtain the spectrum of Higgs radiation emitted from a single kink over the course of one loop oscillationperiod by evaluating Eq. (II.2) with Eq. (D.14), and we find dN (kink)H = ( g H str η ) π ) S (kink) L / | k | / d | k | d Ω ,ψ m H (cid:112) m H L < | k | < M , θ < Θ ( | k | L ) − / (band) (II.8)where the dimensionless coefficient is typically in the range . (cid:46) S (kink) (cid:46) . Here the upper boundon k is M , rather than M √ M L as for the cusp, since the string velocity at the kink is not highly boostedin the loop’s rest frame. The lower bound on k is the same as in the case of the cusp as it arises from ouruse of the saddle point approximation in one of the worldsheet integrals I ± (see Appendix C). Unless theloop is very small, L < M /m H , the lower cutoff will exceed the upper cutoff; in this case, there is noHiggs radiation from the kink within our approximations. This argument is in contrast with the calculationof Ref. [27], where scalar radiation from the kink was also studied.Radiation is emitted into a band that has an angular width Θ( | k | L ) − / and angular length ∼ π .Integrating over the sold angle ∆Ω ≈ π Θ( | k | L ) − / gives the spectrum dN (kink)H = ( g H str η ) π ) Θ S (kink) L / d | k || k | / , ψ m H (cid:112) m H L < | k | < M . (II.9)The total energy emitted by the kink into this channel during one loop oscillation is E (kink) H = (cid:90) ω dN (kink) H = 3( g H str η ) π ) ψ − / Θ S (kink) m H (cid:32) − ψ / mL / M / (cid:33) (II.10)Note that the energy is logarithmically sensitive to both the upper and lower cutoffs of the spectrum. If theloop carries N k kinks, then the average power radiated during one loop oscillation period, T = L/ , isgiven by P (kink) H = Γ (kink) H ( g H str η ) m H L (cid:32) − ψ / m H L / M / (cid:33) (II.11)with Γ (kink) H = π ) N k ψ − / Θ S (kink) . Taking N k ≈ the dimensionless prefactor is estimated to be − (cid:46) Γ (kink) H (cid:46) . This result disagrees with a previous calculation [27] of Higgs radiation from a kink,as explained in Appendix D 2.
3. Higgs Emission from a Kink-Kink Collision
A kink-kink collision occurs when two kinks momentary overlap at the same point on the stringworldsheet. We find the spectrum of Higgs radiation at the collision using Eq. (II.2) along with the scalarintegral in Eq. (D.16): dN (k − k)H = ( g H str η ) π ) S (k − k) ω | k | dω d Ω , m H < ω < M (II.12)where . < S (k − k) < . The bound ω > m H subsumes the bound ω > L − in Eq. (D.16) assuming m H L (cid:29) .The radiation is emitted approximately isotropically (no beaming), and the angular integrationgives dN (k − k)H = ( g H str η ) (2 π ) S (k − k) | k | ω dω , m H < ω < M . (II.13)The total energy emitted by a kink-kink collision is found to be E (k − k) H = (cid:90) ω dN (k − k) H = ( g H str η ) (2 π ) S (k − k) m H . (II.14)Defining N kk as the number of kink-kink collisions during one loop oscillation period, T = L/ , we canexpress the average power radiated by P (k − k) H = Γ (k − k) H ( g H str η ) m H L (II.15)with Γ (k − k) H ≡ π ) N kk S (k − k) . We can estimate the number of collisions per loop oscillation periodas N kk ≈ N k , where N k is the number of kinks on the loop. Estimating N kk ≈ we obtain a range − < Γ (k − k) H < for the dimensionless prefactor. B. Higgs Boson Emission via Quadratic Coupling
The radial component of the Higgs field also couples to the dark string through the quadraticinteraction S HH int = g HH str (cid:90) d σ √− γ φ H ( X ) . (II.16)Unlike in the linear type coupling discussed above, this interaction is not proportional to the Higgs fieldVEV, and it would exist even if the electroweak symmetry were unbroken. This quadratic interaction withthe string produces two Higgs bosons, and thus the final state contains two different momenta, k and ¯ k . Thespectrum of radiation is given by Eq. (B.12) with C = g HH str : dN HH = ( g HH str ) | k | dω d Ω2(2 π ) ¯ | k | d ¯ ω d ¯Ω2(2 π ) (cid:12)(cid:12) I ( k + ¯ k ) (cid:12)(cid:12) (II.17)where k µ = (cid:8) ω , k (cid:9) with ω = ( m H + | k | ) / and m H the Higgs boson mass. The barred quantities aredefined similarly.
1. Higgs-Higgs Emission from a Cusp
Before we can evaluate the spectrum in Eq. (II.17) we must know the value of the scalar integral I ( k + ¯ k ) for a cusp configuration. In Eq. (D.12) we found that this integral evaluates to (cid:12)(cid:12) I (cusp) ( k ) (cid:12)(cid:12) = S (cusp) L / | k | / , ψ m H (cid:112) m H L < | k | , θ < Θ( | k | L ) − / (II.18)when its argument is the approximately null 4-vector momentum k = m H (cid:28) | k | . If the argument ofthe integral is a time-like vector, as in Eq. (II.17), the derivation still leads to Eq. (II.18), but the saddlepoint approximation gives an additional bound on the angle between k and ¯ k . In order to justify the saddlepoint approximation, we were forced to impose the bound in Eq. (C.17). Since the argument of the integralin Eq. (II.17) is k + ¯ k , we must generalize Eq. (C.17) by replacing ω → ω + ¯ ω and | k | → | k + ¯ k | = (cid:113) | k | + ¯ | k | + 2 | k | ¯ | k | cos θ k ¯ k where θ k ¯ k is the angle between k and ¯ k . The bound becomes Θ4 π L / ( ω + ¯ ω − (cid:113) | k | + ¯ | k | + 2 | k | ¯ | k | cos θ k ¯ k ) < ( | k | + ¯ | k | + 2 | k | ¯ | k | cos θ k ¯ k ) / . (II.19)It is useful to consider two limiting cases. If θ k ¯ k = 0 then the inequality translates into a lower bound onthe momentum, ψ m H (cid:112) m H L < ( | k | ¯ | k | ) / (cid:113) | k | + ¯ | k | , (II.20)0and we have used ψ = (Θ / π ) / . When | k | ≈ ¯ | k | we regain the original bound ψ m √ mL < | k | , ¯ | k | . Theinequality also imposes an upper bound on θ k ¯ k . Approximating cos θ k ¯ k ≈ − θ k ¯ k / and using ω ≈ | k | and ¯ ω ≈ ¯ | k | , we can resolve the inequality as θ k ¯ k < ψ − / ( | k | + ¯ | k | ) / L − / (cid:113) | k | ¯ | k | . (II.21)For | k | ≈ ¯ | k | this becomes θ k ¯ k < (2 /ψ ) / ( | k | L ) − / , which agrees with a similar estimate in Ref. [22].From the arguments above, we obtain the cusp integral to be (cid:12)(cid:12) I (cusp) ( k + ¯ k ) (cid:12)(cid:12) = S (cusp) L / ( | k | + ¯ | k | ) / , ψ m H (cid:112) m H L < | k | , ¯ | k | < M √ M L ,θ k ¯ k < ψ − / ( | k | + ¯ | k | ) / L − / (cid:113) | k | ¯ | k | (cone) , θ k +¯ k < Θ( | k | + ¯ | k | ) − / L − / (cone) (II.22)with . (cid:46) S (cusp) (cid:46) . We have also used θ k ¯ k (cid:28) to approximate | k + ¯ k | ≈ | k | + ¯ | k | . InsertingEq. (II.22) into Eq. (II.17) we obtain the spectrum dN (cusp) HH = ( g HH str ) π ) S (cusp) L / | k | ¯ | k | ( | k | + ¯ | k | ) / d | k | d Ω d ¯ | k | d ¯Ω , ψ m H (cid:112) m H L < | k | , ¯ | k | < M √ M L ,θ k ¯ k < ψ − / ( | k | + ¯ | k | ) / L − / (cid:113) | k | ¯ | k | (cone) , θ k +¯ k < Θ( | k | + ¯ | k | ) − / L − / (cone) . (II.23)The upper bound on θ k ¯ k implies that k and ¯ k are approximately parallel to one another, and the upper boundon θ k +¯ k implies that their sum points along the direction of the cusp. The geometry is such that the radiationis emitted into a pair of overlapping cones, and the angular integrations yield (cid:90) d Ω d ¯Ω ≈ (2 π ) ψ − / Θ ( | k | + ¯ | k | ) / | k | ¯ | k | L − / , (II.24)and the spectrum becomes dN (cusp) HH = ( g HH str ) π ) ψ − / Θ S (cusp) d | k | d ¯ | k | ( | k | + ¯ | k | ) , ψ m H (cid:112) m H L < | k | , ¯ | k | < M √ M L . (II.25)The total energy emitted from a cusp is given by E (cusp) HH = (cid:90) ( ω + ¯ ω ) dN (cusp) HH ≈ ( g HH str ) π ) ψ − / Θ S (cusp) M √ M L . (II.26)If the frequency of cusp appearance is f cusp = f c /T with T = L/ is the loop oscillation period, then theaverage power emitted is P (cusp) HH = Γ (cusp) HH ( g HH str M ) √ M L (II.27)1where Γ (cusp) HH ≡ π ) f c ψ − / Θ S (cusp) . Estimating f c ≈ gives − < Γ (cusp) HH < − .Scalar boson pair radiation from a cusp has been calculated previously by Ref. [22]. Our calcula-tion matches the UV-sensitive spectrum, Eq. (II.25), of the earlier reference. In calculating the total power,we integrate up to an energy of M √ M L where /M is the string thickness and √ M L is the boost factorthat translates between the cusp and loop rest frames (see Sec. II A 1). This boost factor was overlookedin the previous calculations, and the power was found to be O ( M/L ) , typically a significant underestimatecompared to Eq. (II.27).
2. Higgs-Higgs Emission from a Kink
We calculate the spectrum of Higgs boson radiation from the kink by evaluating the spectrum inEq. (II.17) using the scalar integral in Eq. (D.14). After also generalizing the saddle point criterion, asdiscussed in Sec. II B 1, we obtain dN (kink) HH = ( g HH str ) π ) S (kink) L / | k | ¯ | k | ( | k | + ¯ | k | ) / d | k | d Ω d ¯ | k | d ¯Ω , ψ m H (cid:112) m H L < | k | , ¯ | k | < M ,θ k ¯ k < ψ − / ( | k | + ¯ | k | ) / L − / (cid:113) | k | ¯ | k | (cone) , θ k +¯ k < Θ ( | k | + ¯ | k | ) − / L − / (band) (II.28)where . < S (kink) < . The momenta k and ¯ k are separated by an angle θ k ¯ k , and their sum is orientedin a band of angular with Θ( | k | + ¯ | k | ) − / L − / . Performing the angular integrations we obtain dN (kink) HH = ( g HH str ) π ) ψ − / Θ S (kink) L / d | k | d ¯ | k | ( | k | + ¯ | k | ) / , ψ m H (cid:112) m H L < | k | , ¯ | k | < M . (II.29)The spectrum is UV-sensitive, which allows us to neglect the lower limit, and upon integrating we find thetotal energy output to be E (kink) HH = (cid:90) ( ω + ¯ ω ) dN (kink) HH ≈ g HH str ) π ) ψ − / Θ S (kink) L / M / (cid:18) − ψm H √ m H LM (cid:19) (II.30)where we have used / [3(2 / − ≈ in the second term. If the loop contains N k kinks, then the averagepower output during one loop oscillation period ( T = L/ ) is given by P (kink) HH = Γ (kink) HH ( g HH str M ) ( M L ) / (cid:18) − ψ m H √ m H LM (cid:19) . (II.31)where Γ (kink) HH ≡ π ) N k ψ − / Θ S (kink) . Estimating N k ≈ and using the range for S (kink) given above,the dimensionless prefactor can be estimated as − < Γ (kink) HH < − .2
3. Higgs-Higgs Emission from a Kink-Kink Collision
To calculate the spectrum of Higgs boson radiation from a kink-kink collision we use the scalarintegral from Eq. (D.16) in the spectrum from Eq. (II.17) to obtain dN (k − k) HH = ( g HH str ) π ) S (k − k) | k | ¯ | k | ( ω + ¯ ω ) dω d Ω d ¯ ω d ¯Ω , m H < ω, ¯ ω < M (II.32)where . < S (k − k) < . The radiation can be emitted isotropically; performing the angular integrationgives a factor of (4 π ) and leaves dN (k − k) HH = ( g HH str ) (2 π ) S (k − k) | k | ¯ | k | ( ω + ¯ ω ) dω d ¯ ω , m H < ω, ¯ ω < M . (II.33)The total energy output of a kink-kink collision is calculated as E (k − k) HH = (cid:90) ( ω + ¯ ω ) dN (k − k) HH = ( g HH str ) π ) S (k − k) M . (II.34)If there are N kk kink-kink collisions during a loop oscillation period T = L/ then the average power isfound to be P (k − k) HH = Γ (k − k) HH ( g HH str M ) M L . (II.35)where Γ (k − k) HH ≡ π ) N kk S (k − k) . For N kk ≈ we can estimate − < Γ (k − k) HH < − using the rangefor S (k − k) given above. C. Z-Boson Emission via Linear Coupling
The interaction S Z int = g Z str (cid:16) ησ (cid:17) (cid:90) dσ µν Z µν ( X ) (II.36)allows Z bosons to be radiated from the string. The radiation calculation is carried out in Appendix B. Thespectrum is given by Eq. (B.21) after replacing C = g Z str ( η/σ ) : dN Z = ( g Z str ) (cid:16) ησ (cid:17) | k | dω d Ω2(2 π ) m Z Π( k ) . (II.37)In this expression ω = ( | k | + m Z ) / with m Z the Z boson mass and Π( k ) is a functional of the string-worldsheet, given by Eq. (B.22). In the following subsections we calculate the spectrum and total power inZ boson emission from cusps, kinks, and kink-kink collisions.3
1. Z Emission from a Cusp
The spectrum of Z boson emission from a cusp is calculated using Eq. (II.37) with the integral inEq. (D.18). Combining these formulae we obtain dN (cusp) Z = ( g Z str ) π ) (cid:16) ησ (cid:17) T (cusp) L / | k | / m Z d | k | d Ω ,ψ m Z (cid:112) m Z L < | k | < M √ M L , θ <
Θ ( | k | L ) − / (cone) . (II.38)where the dimensionless coefficient takes values . (cid:46) T (cusp) (cid:46) . The direction of the outgoing Zboson lies within a cone centered at the cusp and has an opening angle Θ( | k | L ) − / . We integrate over thesolid angle to obtain the spectrum dN (cusp) Z = ( g Z str ) π ) Θ (cid:16) ησ (cid:17) T (cusp) m Z L / | k | / d | k | , ψ m Z (cid:112) m Z L < | k | < M √ M L , (II.39)we integrate over the momentum to obtain the energy output from a single cusp E (cusp) Z = (cid:90) ω dN (cusp) Z = 3( g Z str ) π ) ψ − / Θ (cid:16) ησ (cid:17) T (cusp) m Z (cid:112) m Z L (II.40)and if cusps arise with a frequency f c /T where T = L/ is the loop oscillation period, then the averagepower per loop oscillation is found to be P (cusp) Z = Γ (cusp) Z (cid:16) ησ (cid:17) ( g Z str m Z ) √ m Z L (II.41)where the power coefficient is Γ (cusp) Z ≡ π ) T (cusp) f c ψ − / Θ . Assuming f c ≈ we estimate − (cid:46) Γ (cusp) Z (cid:46) − .
2. Z Emission from a Kink
To calculate the spectrum of Z boson emission from a single kink, we use the expression Eq. (II.37)along with the expression Eq. (D.20) for Π( k ) for a kink to find dN kink Z = ( g Z str ) π ) (cid:16) ησ (cid:17) T (kink) L / | k | / m Z d | k | d Ω ,ψ m Z (cid:112) m Z L < | k | < M , θ < Θ ( | k | L ) − / (band) (II.42)where . < T (kink) < Radiation is emitted in a band with angular width Θ( | k | L ) − / , and weintegrate over the solid angle to find dN kink Z = ( g Z str ) π ) Θ (cid:16) ησ (cid:17) T (kink) L / | k | / m Z d | k | , ψ m Z (cid:112) m Z L < | k | < M (II.43)4The total energy emitted by a kink during one loop oscillation is E (kink) Z = (cid:90) ω dN kink Z = 3( g Z str ) π ) ψ − / Θ (cid:16) ησ (cid:17) T (kink) m Z (cid:32) − ψ / m Z L / M / (cid:33) , (II.44)and if there are N k kinks on the loop then the average radiated power during one loop oscillation period( T = L/ ) is P (kink) Z = Γ (kink) Z (cid:16) ησ (cid:17) ( g Z str m Z ) m Z L (cid:32) − ψ / m Z L / M / (cid:33) (II.45)with Γ (kink) Z = π ) ψ − / Θ T (kink) N k . Estimating N k ≈ gives − < Γ (kink) Z < .
3. Z Emission from a Kink-Kink Collision
Inserting Eq. (D.23) into Eq. (II.37) we obtain the spectrum of Z boson emission from a collisionof kinks to be dN (k − k) Z = ( g Z str ) π ) (cid:16) ησ (cid:17) T (k − k) | k | ω m Z dω d Ω , m Z < ω < M (II.46)with the constant . < T (k − k) < . The emission is isotropic, and after performing the angular integra-tion we obtain dN (k − k) Z ≈ ( g Z str ) π ) (cid:16) ησ (cid:17) T (k − k) | k | ω m Z dω , m Z < ω < M (II.47)The total energy emitted by a kink-kink collision is found to be E (k − k) Z = (cid:90) ω dN (k − k) Z = ( g Z str ) π ) (cid:16) ησ (cid:17) T (k − k) m Z . (II.48)If N kk such collisions occur during one loop oscillation period, T = L/ , then the average power is P (k − k) Z = Γ (k − k) Z (cid:16) ησ (cid:17) ( g Z str m Z ) m Z L (II.49)with Γ (k − k) Z ≡ π ) N kk T (k − k) . Estimating N kk ≈ gives − < Γ (k − k) H < . D. Z Boson Emission via Quadratic Coupling
An interaction of the form S ZZ int = g ZZ str (cid:16) ησ (cid:17) (cid:90) d σ √− γ Z µ ( X ) Z µ ( X ) (II.50)5also allows Z bosons to be radiated from the string. For heavy strings, the coefficient ( η/σ ) is very small,and this radiation channel is negligible. However, we present the calculation of the radiation spectra forcompleteness. The spectrum is given by Eq. (B.28) after replacing C = g ZZ str ( η/σ ) , dN ZZ = 4( g ZZ str ) (cid:16) ησ (cid:17) | k | dω d Ω2(2 π ) ¯ | k | d ¯ ω d ¯Ω2(2 π ) |I ( k + ¯ k ) | , (II.51)where k µ = (cid:8) ω , k (cid:9) and ω = ( m Z + | k | ) / with similar definitions for the barred quantities. Note thesimilarity between Eq. (II.51) and the spectrum of Higgs boson pair radiation given by Eq. (II.17). Sinceboth spectra depend on the same scalar integral, I ( k + ¯ k ) , we can simply carry over all the results fromSec. II B. We need only to make the replacement ( g HH str ) → g ZZ str ) ( η/σ ) .
1. Z-Z Emission from a Cusp
We calculate the spectrum of Z boson radiation from a cusp following Sec. II B 1. We find thespectrum dN (cusp) ZZ = ( g ZZ str ) π ) ψ − / Θ (cid:16) ησ (cid:17) S (cusp) d | k | d ¯ | k | ( | k | + ¯ | k | ) , ψ m Z (cid:112) m Z L < | k | , ¯ | k | < M √ M L , (II.52)the energy radiated per cusp event E (cusp) ZZ = ( g ZZ str ) π ) ψ − / Θ (cid:16) ησ (cid:17) S (cusp) M √ M L , (II.53)and the average power output if cusps arise with frequency f c /LP (cusp) ZZ = Γ (cusp) ZZ (cid:16) ησ (cid:17) ( g ZZ str M ) √ M L . (II.54)The dimensionless coefficient is defined as Γ (cusp) ZZ ≡ π ) f c ψ − / Θ S (cusp) and it may be estimated as − < Γ (cusp) ZZ < − .
2. Z-Z Emission from a Kink
We calculate the spectrum of Z boson radiation from a kink following Sec. II B 2. We find thespectrum dN (kink) ZZ = ( g ZZ str ) π ) ψ − / Θ (cid:16) ησ (cid:17) S (kink) L / d | k | d ¯ | k | ( | k | + ¯ | k | ) / , ψ m Z (cid:112) m Z L < | k | , ¯ | k | < M , (II.55)6the energy radiated per kink during one loop oscillation E (kink) ZZ ≈ g ZZ str ) π ) ψ − / Θ (cid:16) ησ (cid:17) S (kink) L / M / (cid:18) − ψ m Z √ m Z LM (cid:19) , (II.56)and the average power emitted from a loop containing N k kinks P (kink) ZZ = Γ (cusp) ZZ (cid:16) ησ (cid:17) ( g ZZ str M ) ( M L ) / (cid:18) − ψ m Z √ m Z LM (cid:19) . (II.57)The dimensionless coefficient is defined by Γ (kink) ZZ ≡ π ) N k ψ − / Θ S (kink) and it can be estimated as − < Γ (kink) ZZ < − .
3. Z-Z Emission from a Kink-Kink Collision
We calculate the spectrum of Z boson radiation from a collision of two kinks following Sec. II B 2.We find the spectrum dN (k − k) ZZ = ( g ZZ str ) π ) (cid:16) ησ (cid:17) S (k − k) | k | ¯ | k | ( ω + ¯ ω ) dω d ¯ ω , m Z < ω, ¯ ω < M , (II.58)the energy radiated during the collision E (k − k) ZZ = ( g ZZ str ) π ) (cid:16) ησ (cid:17) S (k − k) M , (II.59)and the average power radiated from a loop that experiences N kk collisions during one loop oscillationperiod P (k − k) ZZ = Γ (k − k) ZZ (cid:16) ησ (cid:17) ( g ZZ str M ) M L . (II.60)The dimensionless coefficient is defined by Γ (k − k) ZZ ≡ π ) N kk S (k − k) , and we can estimate − < Γ (k − k) ZZ < − . E. Fermion Emission via Aharonov-Bohm Coupling
The cosmic string can radiate fermions through a direct coupling, such as the ones we have beenstudying for the Higgs and Z bosons, or through a non-local AB interaction. SM fermions couple directlyto the string worldsheet through interactions of the form S ( ψ )int = g ψψ str M (cid:16) ησ (cid:17) (cid:90) d σ √− γ ¯Ψ( X µ )Ψ( X µ ) (II.61)7where g ψψ str is a dimensionless coupling constant, and the factor of ( η/σ ) arises from the mixing betweenthe Higgs field and the HS scalar field [13]. Note that dimensional analysis requires the string mass scale toappear in the denominator. The radiation calculation with S ( ψ )int is very similar to the case of Higgs radiationvia the quadratic interaction, see Sec. B 5. We find the spectrum of ψ radiation to be dN ψψ = 4 (cid:16) ησ (cid:17) (cid:32) g ψψ str g HH str (cid:33) k · ¯ k − m ψ M dN HH (II.62)where dN HH is the spectrum of Higgs radiation, given by Eq. (II.17). Because of the mixing angle factor, ( η/σ ) (cid:28) , this radiation channel is inefficient.The non-local AB interaction provides an additional channel for particle production from the cos-mic string [21]. Refs. [28–30] studied the AB radiation of scalars, fermions, and vectors from a string. Inthese calculations, the authors assumed that the string carries only one kind of magnetic flux, which is usu-ally the case. The structure of the dark string, however, is more complex. As we saw in Ref. [13], the stringcore contains flux of the HS X µ field and the dressing contains flux of the SM Z µ field. When a fermiontravels around the perimeter of the string, outside of both the core and the dressing, its wavefunction picksup an AB phase due to both fluxes, and the overall phase is given by πθ q , where θ q is defined in Eq. (I.6).On the other hand, when the fermion makes a loop around the core by passing through the region of spacecontaining the dressing fields, it will acquire a different AB phase.In order to setup the radiation calculation we must know the effective AB interaction of thefermions with the string. The discussion above is intended to illustrate that this interaction will be scaledependent. At energies below the inverse dressing width, ∼ /η , the core plus dressing can be treatedtogether as a zero width string. In this limit the structure of the string is unimportant, and the AB interactioncan be derived following Refs. [28–30] with the AB phase given by θ q . At higher energies the Comptonwavelength of the radiation drops below the dressing thickness. Here the effective coupling will presumablydecrease as the particle “sees” less and less of the flux carried by the dressing. This behavior is in contrastwith the Higgs and Z boson radiation channels that we considered previously. In those cases, the light SMfields coupled directly to the string core itself, and the dressing was neglected.In light of the discussion above, we will proceed as follows. We calculate the spectrum of radiationdue to the AB interaction where the coupling is set by the AB phase θ q . If the thickness of the string dressingis ∼ /η , then this spectrum is valid up to energies | k | ≈ η √ ηL for the cusp or | k | ≈ η for the kink andkink collision. At higher energies, we suppose that the effective coupling begins to decrease as the fermionradiation begins to penetrate inside of the dressing, and consequently the spectrum drops sharply.The AB interaction can be treated perturbatively as follows. Let V µ ( x ) be the appropriate linear8combination of the X µ and Z µ gauge fields that couples to the fermions, and let g ψ be the coupling constant.Then the interaction is given by L eff = g ψ V µ ( x ) ¯Ψ( x ) γ µ Ψ( x ) . (II.63)We treat V µ as a classical background field induced by the flux that the string carries: Φ = (2 π/g ψ ) θ q . Thislets us write (Lorentz gauge, ∂ µ V µ = 0 ) [21] V µ = − Φ2 (cid:90) ret . d p (2 π ) ip ν p (cid:90) dσ µν e − ip · ( x − X ) (II.64)where the integration contour extends above the poles at p = ± | p | , as in the calculation of a retardedGreen’s function. Note that V µ ( x ) has support outside of the string, unlike the purely local interactions inEqs. (I.4) and (I.5).The interaction inEq. (II.63) allows the string to radiate fermion pairs with momenta k µ = (cid:8) ω = (cid:113) m ψ + | k | , k (cid:9) and ¯ k µ = (cid:8) ¯ ω = (cid:113) m ψ + ¯ | k | , ¯ k (cid:9) . The spectrum is given by Eq. (B.40) after replacing C = − (2 πθ q ) / : dN ψψ = (2 πθ q ) π ) Π( k + ¯ k ) | k | dωd Ω ¯ | k | d ¯ ωd ¯Ω (II.65)where Π is given by Eq. (D.5).
1. Fermion AB Emission from a Cusp
We find the spectrum of radiation from a cusp by inserting Eq. (D.18) into Eq. (II.65): dN (cusp) AB = (2 πθ q ) π ) T (cusp) L / | k | ¯ | k | ( | k | + ¯ | k | ) / d | k | d Ω d ¯ | k | d ¯Ω , ψ m ψ (cid:112) m ψ L < | k | , ¯ | k | < η (cid:112) ηL ,θ k ¯ k < ψ − / ( | k | + ¯ | k | ) / L − / | k | / ¯ | k | / (cone) , θ k +¯ k < Θ( | k | + ¯ | k | ) − / L − / (cone) (II.66)where . (cid:46) T (cusp) (cid:46) . Recall from the discussion of Sec. II B 1 that the momentum sum k + ¯ k isoriented within a cone of angle Θ( | k | + ¯ | k | ) − / L − / centered on the cusp, and the angle between k and ¯ k cannot exceed ψ − / ( | k | + ¯ | k | ) / L − / / (cid:113) | k | ¯ | k | . Upon performing the angular integrations as inEq. (II.24), we obtain dN (cusp) AB = (2 πθ q ) π ) ψ − / Θ T (cusp) d | k | d ¯ | k | ( | k | + ¯ | k | ) , ψ m ψ (cid:112) m ψ L < | k | , ¯ | k | < η (cid:112) ηL (II.67)9We calculate the total energy output as E (cusp) AB = (cid:90) ( ω + ¯ ω ) dN (cusp) AB ≈ (2 πθ q ) π ) ( ψ − / Θ T (cusp) m ψ η (cid:112) ηL (II.68)and the average power output per loop oscillation as P (cusp) AB = Γ (cusp) AB (2 πθ q η ) √ ηL (II.69)where Γ (cusp) AB ≡ π ) ψ − / Θ f c T (cusp) . Using the range for T (cusp) given above, we can estimate − (cid:46) Γ (cusp) AB (cid:46) − .
2. Fermion AB Emission from a Kink
We find the spectrum of radiation from a kink by inserting Eq. (D.20) into Eq. (II.65): dN (kink) AB = (2 πθ q ) π ) T (kink) L / ( | k | + ¯ | k | ) / | k | ¯ | k | d | k | d Ω d ¯ | k | d ¯Ω , ψ m ψ (cid:112) m ψ L < | k | , ¯ | k | < η ,θ k ¯ k < ψ − / ( | k | + ¯ | k | ) / L − / | k | / ¯ | k | / , θ k +¯ k < Θ( | k | + ¯ | k | ) − / L − / (II.70)where . (cid:46) T (kink) (cid:46) . Recall that k + ¯ k is oriented in a ribbon with angular width θ k +¯ k , and theopening angle between k and ¯ k does not exceed θ k ¯ k . After performing the angular integrations we obtain dN (kink) AB = (2 πθ q ) π ) ψ − / Θ T (kink) d | k | d ¯ | k | ( | k | + ¯ | k | ) / L / , ψ m ψ (cid:112) m ψ L < | k | , ¯ | k | < η . (II.71)We calculate the total energy output as E (kink) AB = (cid:90) ( ω + ¯ ω ) dN (kink) AB ≈ πθ q ) π ) ψ − / Θ T (kink) η / L / (cid:32) − ψ / m ψ L / η / (cid:33) , (II.72)and the average power output from N k kinks during one loop oscillation period ( T = L/ ) as P (kink) AB = Γ (kink) AB (2 πθ q η ) ( ηL ) / (cid:32) − ψ / m ψ L / η / (cid:33) (II.73)where Γ (kink) AB ≡ π ) ψ − / Θ T (kink) N k . Using the range for T (kink) given above along with N k ≈ , wecan estimate − (cid:46) Γ (kink) AB (cid:46) .
3. Fermion AB Emission from a Kink-Kink Collision
We find the spectrum radiation from a kink collision by inserting Eq. (D.23) into Eq. (II.65): dN (k − k) AB = (2 πθ q ) π ) T (k − k) ω + ¯ ω ) k + ¯ k ) | k | dωd Ω ¯ | k | d ¯ ωd ¯Ω , m ψ < ω, ¯ ω < η (II.74)0with . < T (k − k) < . In this case, the emission is isotropic, and we can estimate ( k + ¯ k ) ≈ ω ¯ ω up toan O (1) factor associated with the angle between k and ¯ k . The angular integration is trivial, and we find dN (k − k) AB = (2 πθ q ) π ) T (k − k) ω + ¯ ω ) dωd ¯ ω , m ψ < ω, ¯ ω < η . (II.75)We calculate the total energy radiated as E (k − k) AB = (cid:90) ( ω + ¯ ω ) dN (k − k) AB = (2 πθ q ) π ) T (k − k) η , (II.76)and the average power emitted from a loop which experiences N kk collisions during a loop oscillationperiod ( T = L/ ) is found to be P (k − k) AB = Γ (k − k) AB (2 πθ q η ) ηL (II.77)where Γ (k − k) AB ≡ π ) N kk T (k − k) . Using the parameter ranges given above along with N kk ≈ , we canestimate − < Γ (k − k) AB < − . III. SCATTERING CROSS SECTIONS
The interactions discussed in Sec. I allow SM particles to scatter off of the dark string. Interactionsof the Higgs and Z bosons with the string, given by Eqs. (I.4) and (I.5), will lead to a “hard core” scattering,and the AB phases of the SM fermions, given by Eq. (I.6), will lead to a non-local AB scattering. If thecouplings are comparable for the direct and the AB interactions, then the latter generally dominates [19],and therefore we focus on AB scattering here. Moreover, in the cosmological context the dark string willscatter from the SM plasma, which consists mostly of electrons and nuclei at late times.The AB interaction allows fermions to scatter from a cosmic string. The scattering cross section(per length of string) was found to be [21] dσ AB dθ = sin ( πθ q )2 πk ⊥ sin ( θ/ (III.1)where the AB phase for SM fermions, θ q , is given in Eq. (I.6), and k ⊥ is the magnitude of the momentumtransverse to the string. Inserting the expression for θ q and expanding in the θ q (cid:28) limit gives dσ AB dθ ≈ π cos θ W sin (cid:15)g X k ⊥ sin ( θ/ q (III.2)where q is the electromagnetic charge of the fermion.1To study the motion of strings through the cosmological medium, we are interested in the drag(momentum transfer) experienced by the string. This is calculated in terms of a “transport cross section”(see [19]) given by σ AB , t ( k ) = (cid:90) π dθ dσ AB dθ (1 − cos θ ) = 2 k ⊥ sin ( πθ q ) ≈ π cos θ W sin (cid:15)g X q k ⊥ . (III.3)To obtain the total drag due to the entire medium, we must sum over the various species with their respectivecharges q .The derivation of the AB phase, given by Eq. (I.6), assumed that the particle circumnavigates thestring on a length scale larger than the width of the SM dressing. In this way, the particle trajectory enclosesboth the flux carried by the thin HS string core and the thick SM dressing. This length scale is microscopic, ∆ x ∼ η − ≈ − cm , and therefore this assumption is well-justified for the cosmological medium at latetimes, where the inter-particle spacing is much larger than ∆ x . IV. SUMMARY AND CONCLUSION
The dark string couples to the SM fields through the local interactions in Eqs. (I.4) and (I.5)and through the non-local Aharonov-Bohm interactions of charged fermions. These interactions lead toradiation of Higgs bosons, Z bosons, and fermions from cusps, kinks, and kink collisions on cosmic strings.The total power emitted in each of various channels is summarized as follows. For Higgs emission via alinear coupling P (cusp) H = Γ (cusp) H ( g H str η ) √ m H L − < Γ (cusp) H < − (IV.1a) P (kink) H = Γ (kink) H ( g H str η ) m H L (cid:32) − ψ / m H L / M / (cid:33) − < Γ (kink) H < (IV.1b) P (k − k) H = Γ (k − k) H ( g H str η ) m H L − < Γ (k − k) H < , (IV.1c)for Higgs emission via a quadratic coupling P (cusp) HH = Γ (cusp) HH ( g HH str M ) √ M L − < Γ (cusp) HH < − (IV.2a) P (kink) HH = Γ (kink) HH ( g HH str M ) ( M L ) / (cid:18) − ψ m H √ m H LM (cid:19) − < Γ (kink) HH < − (IV.2b) P (k − k) HH = Γ (k − k) HH ( g HH str M ) M L − < Γ (k − k) HH < − , (IV.2c)2for Z boson emission via a linear coupling P (cusp) Z = Γ (cusp) Z (cid:16) ησ (cid:17) ( g Z str m Z ) √ m Z L − < Γ (cusp) Z < − (IV.3a) P (kink) Z = Γ (kink) Z (cid:16) ησ (cid:17) ( g Z str m Z ) m Z L (cid:32) − ψ / m Z L / M / (cid:33) − < Γ (kink) Z < (IV.3b) P (k − k) Z = Γ (k − k) Z (cid:16) ησ (cid:17) ( g Z str m Z ) m Z L − < Γ (k − k) Z < , (IV.3c)for Z boson emission via a quadratic coupling P (cusp) ZZ = Γ (cusp) ZZ (cid:16) ησ (cid:17) ( g ZZ str M ) √ M L − < Γ (cusp) ZZ < − (IV.4a) P (kink) ZZ = Γ (kink) ZZ (cid:16) ησ (cid:17) ( g ZZ str M ) ( M L ) / (cid:18) − ψ m Z √ m Z LM (cid:19) − < Γ (kink) ZZ < − (IV.4b) P (k − k) ZZ = Γ (k − k) ZZ (cid:16) ησ (cid:17) ( g ZZ str M ) M L − < Γ (k − k) ZZ < − , (IV.4c)and for fermion emission via the AB interaction P (cusp) AB = Γ (cusp) AB (2 πθ q η ) √ ηL − < Γ (cusp) AB < − (IV.5a) P (kink) AB = Γ (kink) AB (2 πθ q η ) ( ηL ) / (cid:32) − ψ / m ψ L / η / (cid:33) − < Γ (kink) AB < (IV.5b) P (k − k) AB = Γ (k − k) AB (2 πθ q η ) ηL − < Γ (k − k) AB < − . (IV.5c)Here ψ ≈ . [see Eq. (C.18)] and the other dimensionless coefficients ( Γ factors) depend on undeterminedparameters that characterize the radiating string segment, e.g. , the curvature nearby to the cusp or the sharp-ness of the kink. We quantify our ignorance of these parameters, as described in Appendix D, and this leadsto the ranges shown above. The kink expressions are only valid for small L where the power is positive.Let us highlight the important features of these calculations:1. This system is characterized by three hierarchical length scales, the string thickness, the inverseparticle mass, and the string loop length: /M (cid:28) /m (cid:28) L . The radiation calculation is notamenable to dimensional analysis, because it is always possible to form dimensionless combinationsthat are far from order one, e.g. , M L (cid:29) or m/M (cid:28) . Additionally, some of the spectra are UVsensitive ( dN = d | k | / | k | n with n ≤ ) while others are IR sensitive ( n > ), and as a result someof the power formulae depend on the UV mass scale, M , while others depend on the IR mass scale, η , m H , or m Z .2. In the physically relevant parameter regime, M L (cid:29) m H L (cid:29) , the dominant radiation channel isHiggs emission from cuspy loops via the quadratic interaction, see P (cusp) HH in Eq. (IV.2a).3 (cid:45) (cid:45) m H L P o w e r i n t o H i gg s (cid:64) G e V (cid:68) M (cid:61) (cid:72) (cid:76) GeVG
Μ (cid:61) (cid:72) (cid:45) (cid:76) CuspKinkCollisionHH (cid:72) solid (cid:76) ; H (cid:72) dashed (cid:76)
Gravity (cid:72) dot (cid:45) dashed (cid:76) (cid:45) (cid:45) (cid:45) m H L P o w e r i n t o H i gg s (cid:64) G e V (cid:68) M (cid:61) (cid:72) (cid:76) GeVG
Μ (cid:61) (cid:72) (cid:45) (cid:76) CuspKinkCollisionHH (cid:72) solid (cid:76) ; H (cid:72) dashed (cid:76)
Gravity (cid:72) dot (cid:45) dashed (cid:76) (cid:45) (cid:45) (cid:45) m H L P o w e r i n t o H i gg s (cid:64) G e V (cid:68) M (cid:61) (cid:72) (cid:76) GeVG
Μ (cid:61) (cid:72) (cid:45) (cid:76) CuspKinkCollisionHH (cid:72) solid (cid:76) ; H (cid:72) dashed (cid:76)
Gravity (cid:72) dot (cid:45) dashed (cid:76)
FIG. 2: The total power emitted in Higgs radiation from a cusp (red), a kink (blue), and a kink collision(green) due to the linear (dashed) and quadratic (solid) interactions of the Higgs field with the stringworldsheet. We vary the loop length, L , and show three three different string mass scales M . Forreference, m H L = 10 corresponds to a loop length of L = 1 km , and L = 40 Gly corresponds to m H L = 10 . Note that the scale is different in the left panel.3. The string loop also radiates gravitational waves from cusps, kinks, and kink collisions. The poweroutput into this channel is well-known: P grav = Γ g GM where µ = M is the string tension, Γ g ≈ , and G is Newton’s constant [19]. For comparison, P (cusp) HH ∼ M / /L / . If the stringmass scale is large, then string loops will primarily radiate in the form of gravitational waves, asoriginally observed by Ref. [22]. However, it is important to emphasize that particle emission willdominate if the scale of symmetry breaking is low, e.g. , for a TeV scale string. For instance, tak-ing L ≈
40 Gly to be the size of the horizon today we find P cusp HH /P grav ≈ ( M/ TeV) − / .Moreover, in general Higgs emission dominates over gravitational emission for small loops: L < (Γ (cusp) HH ) ( g HH str ) / (Γ g G M ) .4. Comparing Higgs emission from a cuspy loop via the linear and quadratic interactions, we find P (cusp) HH /P (cusp) H ≈ (Γ (cusp) HH / Γ (cusp) H )( g HH str /g H str ) ( M/m H ) / where we have approximated η ≈ m H .Typically (Γ (cusp) HH / Γ (cusp) H ) ≈ − and ( g HH str /g H str ) ≈ and ( M/m H ) (cid:29) , and we find thatthe quadratic interaction is a much more efficient radiation channel. Note that the dimensionlesscoefficients (the Γ factors) for the quadratic interactions are typically smaller than the correspondingcoefficient for the linear interaction; this is a result of the additional phase space suppression (factorsof π ).5. In Fig. 2 we show the six Higgs boson radiation channels. We use Eqs. (IV.1a) to (IV.2c) taking4 g H str = g HH str = 1 and choosing the largest allowed values for the dimensionless prefactors. In thefirst panel, the line representing gravitational emission is off the scale of the plot at approximately − . For the largest loops, m H L (cid:29) , gravitational emission dominates (pink, dot-dashed). Forthe smallest loops, m H L ≈ , the dominant radiation channel is either pair emission from a cusp(red, solid) or pair emission from a kink (blue, solid). There is no radiation from kinks on largeloops, L (cid:38) M /m H , since the spectrum is bounded as m H √ m H L < | k | < M .6. The spectrum of radiation from kinks extends over the range m √ mL < | k | < M where m isthe particle mass and M is the string mass scale. For momenta below the IR cutoff, a destructiveinterference from different segments of the string loop leads to a suppression of radiation. (In thelanguage of Appendix C 1, the saddle point approximation fails.) For momenta above the UV cutoff,the Compton wavelength of the radiated particle is smaller than the string thickness, /M , and theradiation is once again suppressed. (By comparison, the UV cutoff at a cusp is raised to M √ M L dueto the large boost factor.) Thus only kinks on small loops,
L < M /m , give appreciable radiation.7. The Z boson radiation channels are suppressed compared to the corresponding Higgs radiation chan-nels by the fourth or eight power of ( η/σ ) (cid:28) , and this makes Z boson emission negligible. Thefactor of ( η/σ ) entered the calculation directly in the coupling of the Z boson field to the string, seeEq. (I.4). For the dark string, the Z boson radiation is only possible by virtue of the gauge-kineticmixing, and the mixing angle vanishes in the decoupling limit where ( η/σ ) (cid:28) [13]. For a differentmodel in which this coupling is unsuppressed, the vector boson radiation will be comparable to theHiggs boson radiation, compare Eqs. (IV.2a) and (IV.4a).Throughout this analysis we have assumed that the light SM fields are coupled to the zero thicknessdark string core, which is composed of the heavy HS fields. As we found in Ref. [13], the dark string hasa much richer structure: the thin core is surrounded by a wide dressing made up of the SM Higgs and Zboson fields. The presence of this dressing could lead to a backreaction that was neglected in our particleproduction calculations, and this deserves further investigation. Additionally, as with most calculations ofradiation from cosmic strings, we neglect the more familiar backreaction effect: a reduction in radiationpower as cusps and kinks are gradually smoothed as a result of energy loss in the form of particle andgravitational radiation [31, 32].The particle production calculations that we have presented here play a central role in the study ofastrophysical and cosmological signatures of cosmic strings. For instance, Higgs bosons emitted from thestring at late times will decay and produce cosmic rays that are potentially observable on Earth [24]. In our5followup paper [26], we will study the cosmological evolution of the network of dark strings and assess theprospects for their detection. ACKNOWLEDGMENTS
We are very grateful to Yang Bai, Daniel Chung, Dani`ele Steer, and especially Eray Sabancilar fordiscussions. This work was supported by the Department of Energy at ASU.
Appendix A: Worldsheet Formalism
In this appendix we review the string worldsheet formalism (see, e.g. , [19]). Let τ and σ bethe time-like and space-like worldsheet coordinates, and let X µ ( τ, σ ) be the string worldsheet. Then d σ = dτ dσ is the worldsheet volume element and dσ µν = dτ dσ (cid:15) µναβ (cid:15) ab ∂ a X α ∂ b X β is the worldsheetarea element. Repeated Greek indices are summed from to and Latin indices from to with ∂ X µ = ∂ τ X µ = ˙ X µ and ∂ X µ = ∂ σ X µ = X µ (cid:48) . We define the pullback of the metric as γ ab ≡ g µν ∂ a X µ ∂ b X ν and √− γ ≡ √− det γ = (cid:112) − (1 / (cid:15) ab (cid:15) cd γ ab γ cd .We now specify to the conformal gauge by imposing ˙ X · X (cid:48) = 0 and ˙ X · ˙ X + X (cid:48) · X (cid:48) = 0 . (A.1)Then we have dσ µν = dτ dσ (cid:15) µναβ (cid:16) ˙ X α X (cid:48) β − ˙ X β X (cid:48) α (cid:17) (A.2) √− γ = (cid:12)(cid:12) ˙ X · ˙ X (cid:12)(cid:12) = ˙ X · ˙ X (A.3)where in the last equality we have used the fact that ˙ X µ is not spacelike. We are also free to choose τ = t .Solutions of the equation of motion for a free string, ¨ X = X (cid:48)(cid:48) , can be written as X µ ( t, σ ) = 12 (cid:2) a µ ( σ − ) + b µ ( σ + ) (cid:3) (A.4)where we have introduced the right- and left-movers, a µ ( σ − ) and b µ ( σ + ) , which are functions of σ ± ≡ ( σ ± t ) . For regularly oscillating string loops, these functions obey the periodicity conditions a µ ( L + σ − ) = a µ ( σ − ) and b µ ( L + σ + ) = b µ ( σ + ) (A.5)6in the center of mass frame of the loop. The derivatives are ˙ X ( t, σ ) = 12 (cid:2) − a (cid:48) ( σ − ) + b (cid:48) ( σ + ) (cid:3) X (cid:48) ( t, σ ) = 12 (cid:2) a (cid:48) ( σ − ) + b (cid:48) ( σ + ) (cid:3) (A.6) ¨ X = X (cid:48)(cid:48) = 12 (cid:2) a (cid:48)(cid:48) ( σ − ) + b (cid:48)(cid:48) ( σ + ) (cid:3) . We can use the residual gauge freedom to choose ( a ) µ = (cid:8) − σ − , a ( σ − ) (cid:9) , ( b ) µ = (cid:8) σ + , b ( σ + ) (cid:9) ( a (cid:48) ) µ = (cid:8) − , a (cid:48) ( σ − ) (cid:9) , ( b (cid:48) ) µ = (cid:8) , b (cid:48) ( σ + ) (cid:9) ( a (cid:48)(cid:48) ) µ = (cid:8) , a (cid:48)(cid:48) ( σ − ) (cid:9) , ( b (cid:48)(cid:48) ) µ = (cid:8) , b (cid:48)(cid:48) ( σ + ) (cid:9) , (A.7)along with the condition that a (cid:48) and b (cid:48) should be null, which implies | a (cid:48) ( σ − ) | = | b (cid:48) ( σ + ) | = 1 a (cid:48) · a (cid:48)(cid:48) = b (cid:48) · b (cid:48)(cid:48) = a (cid:48) · a (cid:48)(cid:48) = b (cid:48) · b (cid:48)(cid:48) = 0 (A.8) a (cid:48) · a (cid:48)(cid:48)(cid:48) + a (cid:48)(cid:48) · a (cid:48)(cid:48) = b (cid:48) · b (cid:48)(cid:48)(cid:48) + b (cid:48)(cid:48) · b (cid:48)(cid:48) = 0 . This parametrization lets us write d σ = dτ dσ = 12 dσ + dσ − √− γ = − a (cid:48) · b (cid:48) (A.9) dσ µν = dτ dσ (cid:15) µναβ b (cid:48) α a (cid:48) β . Note that a (cid:48) · b (cid:48) = − − a (cid:48) · b (cid:48) ≤ and therefore √− γ ≥ as it should be. Appendix B: Calculation of Particle Radiation from the String
In this appendix, we calculate the spectrum of scalar and vector boson emission due to a couplingwith a cosmic string of the linear or quadratic form. We also derive the spectrum of fermions emitted due toa direct coupling and an Aharonov-Bohm coupling. The results we obtain are not unique to the dark stringmodel; they apply to any model that has couplings of the form considered here.We use the matrix element formalism to perform these calculations [22]. Since the linear couplinggives rise to a classical source for the scalar or vector field, the radiation in these cases can also be calculatedby solving the classical field equation [24, 25]. We have verified that both approaches give identical spectra.We also retain all factors of and π , which were neglected in the previous calculations.7
1. Scalar Radiation via Linear Coupling
Consider a real scalar field φ ( x ) of mass m that is coupled to the string worldsheet X µ ( τ, σ ) through the effective interaction L eff = A φ ( x ) (cid:90) d σ √− γ δ (4) ( x − X ) (B.1)where A is an arbitrary real parameter with mass dimension one. We calculate the amplitude for particleproduction by making a perturbative expansion in A . Then to leading order we have A = i (cid:90) d x (cid:10) k (cid:12)(cid:12) L eff ( x ) (cid:12)(cid:12) (cid:11) (B.2)where (cid:12)(cid:12) k (cid:11) is a one-particle state of momentum k . The action of the field operator on the one-particle stateis simply φ ( x ) (cid:12)(cid:12) k (cid:11) = e − ik · x (cid:12)(cid:12) (cid:11) and (cid:10) k (cid:12)(cid:12) φ ( x ) = e ik · x (cid:10) (cid:12)(cid:12) (B.3)where k µ = (cid:8) ω, k (cid:9) with ω = (cid:112) m + | k | . Then upon inserting Eq. (B.1) into Eq. (B.2) we obtain A ( k ) = i A (cid:90) d x e ik · x (cid:90) d σ √− γ δ (4) ( x − X ) = i A I ( k ) (B.4)where I ( k ) ≡ (cid:90) d σ √− γ e ik · X . (B.5)In Appendix D we calculate this integral for various string configurations, as specified by X µ ( τ, σ ) .For a given X we calculate the number of scalar bosons emitted into a phase space volume d k = | k | d | k | d Ω as dN = d k (2 π ) ω |A ( k ) | . (B.6)Using Eq. (B.4) in Eq. (B.6) we obtain the final spectrum dN = A d k (2 π ) ω |I ( k ) | (B.7)where the dimensionful coefficient is equal to A = g H str η for the dark string. More accurately, the initial state is not vacuum, but it is a state containing the string, (cid:12)(cid:12) S (cid:11) , and the final state contains a deforma-tion of the initial string state, (cid:12)(cid:12) S (cid:48) (cid:11) . Provided that the radiation has a negligible backreaction on the string state, one can neglectthe deformation and then (cid:10) S (cid:48) (cid:12)(cid:12) S (cid:11) ≈ (cid:10) S (cid:12)(cid:12) S (cid:11) = (cid:10) (cid:12)(cid:12) (cid:11) [22].
2. Scalar Radiation via Quadratic Coupling
Consider a real scalar field φ ( x ) of mass m that is coupled to the string worldsheet X µ ( τ, σ ) through the effective interaction L eff = C φ ( x ) (cid:90) d σ √− γ δ (4) ( x − X ) (B.8)where C is an arbitrary real parameter with mass dimension zero. We can calculate the radiation of scalarboson pairs using perturbation theory provided that C (cid:28) . Consider the radiation of a boson pairwith momenta k and ¯ k . We can introduce the 4-vectors k µ = (cid:8) ω = √ k + m , k (cid:9) and ¯ k µ = (cid:8) ¯ ω = (cid:112) ¯ k + m , ¯ k (cid:9) . To leading order in C the amplitude for this process is A = i (cid:90) d x (cid:10) k ¯ k (cid:12)(cid:12) L eff ( x ) (cid:12)(cid:12) (cid:11) . (B.9)Inserting Eq. (B.8) into Eq. (B.9) and using Eq. (B.3) we obtain A = iC I ( k + ¯ k ) (B.10)where I ( k ) was defined in Eq. (B.5). The number of scalar bosons emitted into the phase space volume d k d ¯ k = | k | d | k | d Ω ¯ | k | d ¯ | k | d ¯Ω is calculated as dN = d k (2 π ) ω d ¯ k (2 π ) ω |A| . (B.11)Using Eq. (B.10) this becomes dN = C d k (2 π ) ω d ¯ k (2 π ) ω (cid:12)(cid:12) I ( k + ¯ k ) (cid:12)(cid:12) (B.12)where C = g HH str for the dark string.
3. Vector Radiation via Linear Coupling
Consider a vector field A µ ( x ) of mass m that couples to the string worldsheet X µ ( τ, σ ) , via thelinear interaction L eff = C F µν ( x ) (cid:90) dσ µν δ (4) ( x − X ) (B.13)where F µν = ∂ µ A ν − ∂ ν A µ is the field strength tensor and C is a real parameter of mass dimension zero.Recall that the worldsheet area element was defined in Eq. (A.2). Since the radiation will be relativistic, we9can treat the gauge boson as transversely polarized with two allowed helicties λ = ± . We calculate theamplitude to radiate a vector boson with momentum k and helicity λ as A = i (cid:90) d x (cid:10) k , λ (cid:12)(cid:12) L eff ( x ) (cid:12)(cid:12) (cid:11) . (B.14)The action of the field operator on the one-particle state is A ν ( x ) (cid:12)(cid:12) k , λ (cid:11) = (cid:15) ν ( k, λ ) e − ik · x (cid:12)(cid:12) (cid:11) and (cid:10) k , λ (cid:12)(cid:12) A ν ( x ) = (cid:15) ∗ ν ( k, λ ) e ik · x (cid:10) (cid:12)(cid:12) (B.15)where k µ = (cid:8) ω = (cid:112) m + | k | , k (cid:9) . Inserting Eq. (B.13) into Eq. (B.14) and using Eq. (B.15) gives A = − C k µ (cid:15) ∗ ν ( k, λ ) I µν ( k ) (B.16)where I µν ( k ) ≡ (cid:90) dσ µν e ik · X . (B.17)Then the number of vector bosons emitted into the phase space volume d k = | k | d | k | d Ω is calculated as dN = (cid:88) λ d k (2 π ) ω |A| (B.18)where we sum over the two polarization states. Using Eq. (B.16) this becomes dN = C d k (2 π ) ω k µ k α I µν ( k ) I αβ ( k ) ∗ (cid:88) λ (cid:15) β ( k, λ ) (cid:15) ∗ ν ( k, λ ) . (B.19)We perform the spin sum using the completeness relationship (cid:88) λ = ± (cid:15) β ( k, λ ) (cid:15) ∗ ν ( k, λ ) = − g βν . (B.20)Doing so we find the spectrum to be dN = C d k (2 π ) ω k Π( k ) (B.21)where the (positive, real) function Π( q ) ≡ − g νβ q µ q α q I µν ( q ) I αβ ( q ) ∗ (B.22)has dimensions of length and carries the dependence on the string worldsheet. By choosing C = g Z str ( η/σ ) we obtain the spectrum of Z boson radiation from the dark string.0
4. Vector Radiation via Quadratic Coupling
Consider a vector field A µ of mass m that couples to the string worldsheet via the quadratic inter-action L eff = C A µ ( x ) A µ ( x ) (cid:90) d σ √− γ δ (4) ( x − X ) (B.23)where C is a real parameter of mass dimension zero. The amplitude to radiate a pair of vector bosons withmomenta k and ¯ k and helicities λ and ¯ λ is calculated as A = i (cid:90) d x (cid:10) k , λ ; ¯ k , ¯ λ (cid:12)(cid:12) L eff ( x ) (cid:12)(cid:12) (cid:11) . (B.24)We can introduce the 4-vectors k µ = (cid:8) ω = √ k + m , k (cid:9) and ¯ k µ = (cid:8) ¯ ω = (cid:112) ¯ k + m , ¯ k (cid:9) . Uponinserting Eq. (B.23) into Eq. (B.24) and using Eq. (B.15) we obtain A = iC (cid:15) ∗ µ ( k, s ) (cid:15) ∗ ν (¯ k, ¯ s ) g µν I ( k + ¯ k ) (B.25)where I ( k ) was defined in Eq. (B.5). Then the number of vector bosons emitted into the phase space volume d k d ¯ k = | k | d | k | d Ω ¯ | k | d ¯ | k | d ¯Ω is calculated as dN = (cid:88) λ (cid:88) ¯ λ d k (2 π ) ω d ¯ k (2 π ) ω |A| (B.26)where we sum over the transverse polarization states λ, ¯ λ = ± . (Since the radiation is highly boosted, wecan neglect the longitudinal polarization states.) Using Eq. (B.25) this becomes dN = C d k (2 π ) ω d ¯ k (2 π ) ω |I ( k + ¯ k ) | g µν g αβ (cid:88) λ (cid:15) α ( k, λ ) (cid:15) ∗ µ ( k, λ ) (cid:88) λ (cid:15) β (¯ k, ¯ λ ) (cid:15) ∗ ν (¯ k, ¯ λ ) . (B.27)We evaluate the spin sums using the completeness relation in Eq. (B.20) to find dN = 4 C d k (2 π ) ω d ¯ k (2 π ) ω |I ( k + ¯ k ) | . (B.28)For the dark string model we take C = g ZZ str ( η/σ ) .
5. Dirac Spinor Radiation – Direct Coupling
Consider a Dirac field Ψ( x ) of mass m that is coupled to the string worldsheet X µ ( τ, σ ) throughthe effective interaction L eff = CM ¯Ψ( x )Ψ( x ) (cid:90) d σ √− γ δ (4) ( x − X ) (B.29)1where C is an arbitrary real parameter with mass dimension zero, and M is the string mass scale. Considerthe radiation of a particle / anti-particle pair with momenta k and ¯ k and spins s and ¯ s . We can introduce the4-vectors k µ = (cid:8) ω = √ k + m , k (cid:9) and ¯ k µ = (cid:8) ¯ ω = (cid:112) ¯ k + m , ¯ k (cid:9) . To leading order the amplitudefor this process is A = i (cid:90) d x (cid:10) k , s ; ¯ k , ¯ s (cid:12)(cid:12) L eff ( x ) (cid:12)(cid:12) (cid:11) . (B.30)The action of the field operator on the one-particle state is given by (cid:10) k , s (cid:12)(cid:12) ¯Ψ( x ) = ¯ u ( k , s ) e ik · x (cid:10) (cid:12)(cid:12) and (cid:10) ¯ k , ¯ s (cid:12)(cid:12) Ψ( x ) = v (¯ k , ¯ s ) e i ¯ k · x (cid:10) (cid:12)(cid:12) . (B.31)Inserting Eq. (B.29) into Eq. (B.30) we obtain A = i CM ¯ u ( k , s ) v (¯ k , ¯ s ) I ( k + ¯ k ) (B.32)where I ( k ) was defined in Eq. (B.5). The number of particle pairs emitted into the phase space volume d k d ¯ k = | k | d | k | d Ω ¯ | k | d ¯ | k | d ¯Ω is calculated as in Eq. (B.26) where now the sum is over spin states s, ¯ s = ± / . We use the completeness relations, (cid:88) s u ( k , s )¯ u ( k , s ) = ( k µ γ µ + m ) and (cid:88) ¯ s v (¯ k , ¯ s )¯ v (¯ k , ¯ s ) = (¯ k µ γ µ − m ) . (B.33)Using the familiar Dirac gamma trace relations, we obtain dN = 4 C d k (2 π ) ω d ¯ k (2 π ) ω (cid:12)(cid:12) I ( k + ¯ k ) (cid:12)(cid:12) k · ¯ k − m M (B.34)where C = g ψψ str ( η/σ ) for the dark string.
6. Dirac Spinor Radiation – AB Coupling
Consider a Dirac field Ψ( x ) of mass m that is coupled to the string worldsheet X µ ( τ, σ ) throughthe effective interaction L eff = C ¯Ψ( x ) γ µ Ψ( x ) (cid:90) ret . d p (2 π ) ip ν p I µν ( p ) e − ip · x (B.35)where C (cid:28) is an arbitrary real parameter with mass dimension zero, and I µν ( k ) was defined inEq. (B.17). In the momentum integral, the integration contour is extended above both poles at p = ± | p | .Following Sec. B 5 we calculate the amplitude for the radiation of a particle/anti-particle pair: A = − C ¯ u ( k , s ) γ µ v (¯ k , ¯ s ) q ν q I µν ( q ) (B.36)2where q ≡ k + ¯ k . The number of particle pairs emitted into the phase space volume d k d ¯ k = | k | d | k | d Ω ¯ | k | d ¯ | k | d ¯Ω is calculated as in Eq. (B.26) where now the sum is over spin states s, ¯ s = ± / .Using Eq. (B.36) we obtain dN = 4 C d k (2 π ) ω d ¯ k (2 π ) ω (cid:0) k µ ¯ k α + k α ¯ k µ − ( m + k · ¯ k ) g µα (cid:1) q ν q β q I µν ( q ) I αβ ( q ) ∗ . (B.37)Then using the antisymmetry of I µν we find dN = 2 C d k (2 π ) ω d ¯ k (2 π ) ω (cid:104) Π( k + ¯ k ) − | Υ( k, ¯ k ) | (cid:105) (B.38)where Π( q ) is defined in Eq. (B.22) and Υ( k, ¯ k ) ≡ k µ ¯ k ν ( k + ¯ k ) I µν ( k + ¯ k ) . (B.39)In general, the evaluation of Eq. (B.38) is very involved and must be done numerically for some choice ofloops as in [29]. However, to extract the radiation spectrum, it is sufficient to note that dN > , and sothe term containing Υ is never larger than the term containing Π [see also Eq. (D.10)]. Hence, to extractscalings, we will take dN ≈ C d k (2 π ) ω d ¯ k (2 π ) ω Π( k + ¯ k ) (B.40)where for the dark string C = − (2 πθ q ) / . Appendix C: Calculation of the Worldsheet Integrals
In Appendix B we encountered the two integrals I ( k ) = (cid:90) d σ √− γ e ik · X (C.1) I µν ( k ) = (cid:90) d σ µν e ik · X (C.2)while calculating the radiation spectra. In this appendix and the next, we will analytically calculate theseintegrals for the cusp, kink, and kink-kink collision string configurations.It is convenient to define the integrals I µ + ( b ; k ) ≡ (cid:90) L dσ + b (cid:48) µ e ik · b/ and I µ − ( a ; k ) ≡ (cid:90) L dσ − a (cid:48) µ e ik · a/ (C.3) There is a danger that there can be cancellations between the Π and | Υ | terms but we find that our scalings agree with thebehavior that was numerically obtained in [29] for similar loops. I + is a functional of b µ ( σ + ) with parameter k µ , and similarly I − is a functional of a µ ( σ − ) . For aregularly oscillating string loop, the periodicity of a (cid:48) and b (cid:48) implies the identities k · I ± = 0 . (C.4)Additionally, for such a loop we can factorize the original integrals from Eqs. (C.1) and (C.2) in terms of I + and I − . We use Eq. (A.9) to factor the integrands, and we use the periodicity of the loop oscillationto rewrite the domain of integration as (cid:82) L dσ (cid:82) T dτ = (1 / (cid:82) L dσ + (cid:82) L dσ − where T = L/ is the looposcillation period. Doing so gives I ( k ) = − g αβ ( I + ( b ; k )) α ( I − ( a ; k )) β (C.5) I µν ( k ) = 12 (cid:15) µναβ ( I + ( b ; k )) α ( I − ( a ; k )) β . (C.6)The problem is now reduced to calculating the two integrals, I µ + and I µ − , for a given loop configuration,specified by a µ and b µ .These integrals cannot be performed analytically for general configurations. We, therefore, focuson the configurations that we expect to maximize the integrals, since this corresponds to maximum particleradiation. It turns out that for these optimum configurations, the saddle point and the discontinuity, theintegrals are analytically tractable.
1. Saddle Point Integral
The integrals in Eq. (C.3) become analytically tractable if there is a saddle point at which thephase is stationary [33]. For the sake of discussion consider the integral I + . Its phase can be expandedabout σ + = σ s as k · b ( σ + )2 = k · b s k · b (cid:48) s σ + − σ s ) + k · b (cid:48)(cid:48) s σ + − σ s ) + k · b (cid:48)(cid:48)(cid:48) s
12 ( σ + − σ s ) + . . . . (C.7)Subscripts are used to denote evaluation of the function at a particular point, e.g. , b (cid:48) s = b (cid:48) ( σ s ) . We say that σ s is a saddle point if the stationary phase criterion, k · b (cid:48) s = 0 , (C.8)is satisfied. Using Eq. (A.7) this can be written as k · b (cid:48) s = ω − k · b (cid:48) s = ω − | k | cos θ (C.9)4where θ is the angle between k and b (cid:48) s . If the particle being radiated is massless, ω = | k | , then the saddlepoint criterion is satisfied by choosing k = | k | b (cid:48) s ( i.e. , θ = 0 ). Then it follows from the identity in Eq. (A.8)that k · b (cid:48)(cid:48) s = 0 as well, and the leading term in Eq. (C.7) is cubic.For massive particle radiation the saddle point criterion cannot be satisfied exactly. Instead, wehave instead a quasi-saddle point, σ + = σ qsp , at which the phase is approximately stationary: k = | k | b (cid:48) qsp , k · b (cid:48) qsp = ω − | k | , k · b (cid:48)(cid:48) qsp = 0 , k · b (cid:48)(cid:48)(cid:48) qsp = | k | | b (cid:48)(cid:48) qsp | , (C.10)where we have used Eq. (A.8). It will be convenient to write b (cid:48)(cid:48) qsp = 2 πL β qsp ˆ b (cid:48)(cid:48) qsp and a (cid:48)(cid:48) qsp = 2 πL α qsp ˆ a (cid:48)(cid:48) qsp (C.11)where the hatted quantities are unit vectors. The dimensionless parameters α asp and β asp are related tothe acceleration or curvature of the loop at the quasi-saddle point (recall Eq. (A.6)). The stationary phaseapproximation is still applicable as long as ( k · b (cid:48) qsp )( σ + ) (cid:28) ( k · b qsp ) (cid:48)(cid:48)(cid:48) ( σ + ) (cid:28) π .Suppose that we are given a configuration b µ ( σ + ) and a k µ such that there exists some point σ + = σ s where the quasi-saddle point condition, Eq. (C.10), is satisfied. Then the integral from Eq. (C.3)can be approximated by expanding in ∆ σ = σ + − σ s , which gives I µ + ( b ; k ) ≈ e i k · bs (cid:90) L − σ s − σ s d (∆ σ ) (cid:2) ( b (cid:48) s ) µ + ( b (cid:48)(cid:48) s ) µ ∆ σ (cid:3) exp [ iφ (∆ σ )] . (C.12)The phase is also expanded in powers of ∆ σ/L as φ (∆ σ ) = φ (∆ σ ) + φ (∆ σ ) + . . . where φ ≡ ω − | k | σ and φ ≡ π | k | L (∆ σ ) . (C.13)Here we have introduced the dimensionless parameter Θ ≡ (6 /πβ s ) / , and the shape parameter is β s = L | b (cid:48)(cid:48) s | / (2 π ) as per Eq. (C.11).As long as φ is negligible, the integral is in the stationary phase regime, and it can be evaluateddirectly with the saddle point approximation. Since the integral is dominated by the saddle point, we canextend the limits of integration to infinity. Doing so we obtain I + ( b ; k ) ≈ e i k · bs (cid:90) ∞−∞ d (∆ σ ) (cid:2) b (cid:48) s + b (cid:48)(cid:48) s ∆ σ (cid:3) exp [ iφ (∆ σ )]= e i k · bs L (cid:18) A + b (cid:48) s ( | k | L ) / + iB + Lb (cid:48)(cid:48) s ( | k | L ) / (cid:19) (C.14)where A + = (2 π ) / /
3) Θ and B + = Γ(2 / √ (2 π ) / , (C.15)5and Θ = (6 /πβ s ) / was defined in the paragraph above.The linear phase, φ , must be negligible if the saddle point approximation is to be valid. We definethe “width of the saddle point” by the condition φ (∆ σ max ) = 2 π , which gives ∆ σ max = Θ L ( | k | L ) − / . (C.16)Imposing φ (∆ σ max ) < φ (∆ σ max ) leads to the bound Θ4 π L / ( ω − | k | ) < | k | / . (C.17)The left-hand side vanishes in the relativistic limit, and the bound becomes saturated as the momentum islowered. Approximating ω ≈ | k | + m / | k | we obtain a lower bound on the momentum [22] ψ m √ mL < | k | with ψ = (cid:18) Θ8 π (cid:19) / = 3 / π √ β s . (C.18)We can also translate ∆ σ max into an upper bound on the angle between k and b (cid:48) s : θ max = ∆ σ max L = Θ ( | k | L ) − / . (C.19)For the I − integral, the analysis is similar, but the saddle point criterion is replaced with k · a (cid:48) s = − ω − k · a (cid:48) s = 0 implying that k = −| k | a (cid:48) s at the quasi-saddle point. Consequently, in the equationsanalogous to Eq. (C.10) all the signs on the right hand side are flipped. The results for both integrals can besummarized as I + ≈ A + L b (cid:48) s ( | k | L ) / + iB + L b (cid:48)(cid:48) s ( | k | L ) / , ψ m √ mL < | k | , θ kb (cid:48) s < Θ ( | k | L ) − / I − ≈ A − L a (cid:48) s ( | k | L ) / + iB − L a (cid:48)(cid:48) s ( | k | L ) / , ψ m √ mL < | k | , θ ka (cid:48) s < Θ ( | k | L ) − / (C.20)where θ kb (cid:48) s and θ ka (cid:48) s are the angles between k and b (cid:48) s or a (cid:48) s , respectively. The dimensionless parameters aredefined as A ± = 2 π / (cid:18) π γ ± (cid:19) / and B ± = ± Γ(2 / √ (cid:18) π γ ± (cid:19) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ + = β s γ − = α s , (C.21)and the dimensionless shape parameters, β s and α s , are defined as in Eq. (C.11). For shorter wavelengthradiation, | k | < ψ m √ mL , there is no saddle point, and the integral vanishes rapidly. Also note that theapproximations to I ± in Eq. (C.20) satisfy the identities in Eq. (C.4) for k = | k | c (cid:48)± up to O ( m / | k | ) terms.6
2. Discontinuity Integral
In this appendix we will evaluate the integrals in Eq. (C.3) for the case in which the gradient ofthe string worldsheet, ∂ σ X µ , has a discontinuity [33]. We first suppose that b µ ( σ + ) has N k discontinuitiescorresponding to N k kinks on the string loop. The typical distance between the kinks will be D = L/N k .To calculate the contribution to I + coming from a single discontinuity located at σ + = σ d we parametrize b (cid:48) µ ( σ + ) = b (cid:48) µ + = (cid:8) , ˆ m + (cid:9) < σ + − σ d < D/ b (cid:48) µ − = (cid:8) , ˆ m − (cid:9) − D/ < σ + − σ d < (C.22)where ˆ m ± are unit vectors and b ± = (cid:8) σ + , ( σ + − σ d ) ˆ m ± (cid:9) . Inserting Eq. (C.22) into Eq. (C.3) weapproximate the worldsheet integral as I + ≈ (cid:90) D/ − D/ dσ + b (cid:48) ( σ + ) e ik · b (cid:48) σ + / ≈ (cid:34) ω (cid:32) b (cid:48) + ˆ k · b (cid:48) + − b (cid:48)− ˆ k · b (cid:48)− (cid:33) − ω (cid:32) b (cid:48) + ˆ k · b (cid:48) + e i (ˆ k · b (cid:48) + ) ωD − b (cid:48)− ˆ k · b (cid:48)− e − i (ˆ k · b (cid:48)− ) ωD (cid:33)(cid:35) e i ( ωσd + π ) (C.23)where ˆ k µ ≡ k µ /ω = (cid:8) , k /ω (cid:9) . Upon integrating over the entire loop, the second term cancels among thecontributions from different discontinuities (summing all kinks). Then we can drop both this second termand the overall phase to write the contribution from a single discontinuity as I + ≈ ω (cid:16) β + b (cid:48) + − β − b (cid:48)− (cid:17) (C.24)with β ± ≡ / (ˆ k · b (cid:48)± ) = 2 / (1 − ˆ k · ˆ m ± ) .To calculate the integral I − we parametrize a (cid:48) ( σ − ) in terms of a (cid:48)± in analogy with Eq. (C.22). Wecan summarize the results of both calculations as follows I + ( k ) ≈ ω (cid:16) β + b (cid:48) + − β − b (cid:48)− (cid:17) , β ± = 2ˆ k · b (cid:48)± , L − < ω I − ( k ) ≈ ω (cid:16) α + a (cid:48) + − α − a (cid:48)− (cid:17) , α ± = 2ˆ k · a (cid:48)± , L − < ω . (C.25)The dimensionless coefficients are bounded as ≤ β ± , α ± . In the limit that k coincides with one of thediscontinuity vectors, b (cid:48)± or a (cid:48)± , one finds that β ± or α ± → ∞ . This apparent divergence is an artifact ofneglecting the second set of terms in Eq. (C.23), and upon retaining these terms one can see that I + ∼ D (cid:28) /ω in the limit that (ˆ k · b (cid:48) + ) ωD (cid:28) . Therefore we must restrict ourselves to the regime ω > D − ∼ N k L − and where k · b (cid:48)± is away from zero; it follows that ≤ β ± , α ± (cid:46) few . To properly treat the case k · b (cid:48) + = 0 in which the phase is stationary, one should use the saddle point approximation, as described inSec. C 1.7 Appendix D: Scalar and Tensor Integrals for Cusps, Kinks, and Kink Collisions
Here we evaluate the scalar and tensor integrals, I and I µν given by Eqs. (C.5) and (C.6), forthe cusp, kink, and kink-kink collision string configurations. For the scalar integral, we will only be inter-ested in the modulus |I| . For the tensor integral, we will only be interested in the (positive, real) scalarcombinations Π( q ) = − g νβ q µ q α q I µν ( q ) I αβ ( q ) ∗ (D.1) Υ( k, ¯ k ) = 2 k µ ¯ k ν ( k + ¯ k ) I µν ( k + ¯ k ) (D.2)which were originally defined in Eqs. (B.22) and (B.39).We can simply the expression for Π( q ) as follows. Using Eq. (C.6) and the identity ( − g νβ ) (cid:15) µνγδ (cid:15) αβρσ = g µα g γρ g δσ + g µρ g γσ g δα + g µσ g γα g δρ − g µα g γσ g δρ − g µρ g γα g δσ − g µσ g γρ g δα (D.3)we can write Π as Π( q ) = 14 (cid:104) ( I + · I ∗ + )( I − · I ∗− ) − |I + · I ∗− | (cid:105) + 12 1 q Re (cid:104) ( q · I ∗ + )( q · I − )( I + · I ∗− ) (cid:105) −
14 1 q (cid:104) ( q · I + )( q · I ∗ + )( I − · I ∗− ) + ( q · I − )( q · I ∗− )( I + · I ∗ + ) (cid:105) . (D.4)Furthermore, from the periodicity of the string worldsheet, we have the identity q ·I ± ( q ) = 0 [see Eq. (C.4)].Making this simplification we finally obtain Π( q ) = 14 (cid:104) ( I + · I ∗ + )( I − · I ∗− ) − |I + · I ∗− | (cid:105) . (D.5)We can simplify the expression for Υ( k, ¯ k ) as follows. Let q = k + ¯ k and p = k − ¯ k . UsingEq. (C.6) we can express Υ as k, ¯ k ) q = p µ q ν I + α ( q ) I − β ( q ) (cid:15) µναβ = − p q · ( I + × I − ) + q p · ( I + × I − ) − I p · ( q × I − ) + I − p · ( q × I + ) . (D.6)Using the identities q · p = q · I + = q · I − = 0 , this can also be written as [29] Υ( k, ¯ k ) = 12 q p · ( I + × I − ) . (D.7)8To compare Υ with Π , it is convenient to move to the frame in which q µ = (cid:8) q , (cid:9) . Then theidentities q · p = q · I + = q · I − = 0 require p , I + , and I − to have vanishing time-like components. In thisframe, we can write Π( q ) = | I + | | I − | ( θ + − ) (D.8)where θ + − is the angle between I + and I − . Further denoting θ p + − as the angle between p and I + × I − we have | Υ( k, ¯ k ) | = (cid:34)(cid:18) | p | q (cid:19) cos ( θ p + − ) (cid:35) | I + | | I − | ( θ + − ) . (D.9)The two expressions are related by | Υ( k, ¯ k ) | = (cid:34)(cid:18) | k − ¯ k | ω + ¯ ω (cid:19) cos ( θ p + − ) (cid:35) Π( q ) (D.10)where the quantity is square brackets is always ≤ . The inequality is saturated when p is aligned with I + × I − ( i.e. , θ p + − ≈ ) and either | k | (cid:29) ¯ | k | or ¯ | k | (cid:29) | k | .
1. Scalar Integral – Cusp
A cusp occurs when both integrals I + ( b ; q ) and I − ( a ; q ) have a saddle point at the same value of q µ [see Eq. (C.10)]. This requires q = | q | b (cid:48) c = −| q | a (cid:48) c or equivalently a (cid:48) c = − b (cid:48) c . (D.11)The scalar integral, I from Eq. (C.5), is evaluated using the expressions for I ± in Eq. (C.20). Using theidentities from Eq. (A.8) most of the four-vector contractions vanish. The surviving term is proportionalto a (cid:48)(cid:48) c · b (cid:48)(cid:48) c = (2 π/L ) α c β c ˆ a (cid:48)(cid:48) c · ˆ b (cid:48)(cid:48) c where we have used shape shape parameters, introduced in Eq. (C.11).Then, the squared integral evaluates to (cid:12)(cid:12) I (cusp) ( q ) (cid:12)(cid:12) = S (cusp) L / | q | / , ψ m √ mL < | q | , θ < Θ ( | q | L ) − / (cone) (D.12)where S (cusp) ≡ π α c β c B − B cos θ ab with B ± defined in Eq. (C.21), and where θ ab is the angle between a (cid:48)(cid:48) c and b (cid:48)(cid:48) c . The angle between k and b (cid:48) c = − a (cid:48) c is bounded above by the saddle point criterion, and therefore k falls within a cone of opening angle Θ( | q | L ) − / centered on the cusp.The dimensionless prefactor, S (cusp) , may be estimated using the expressions for B ± in Eq. (C.21).The shape parameters, α c and β c , are expected to be O (1) , but their precise values cannot be determined9without greater knowledge of the nature of the cusp. In order to track how this uncertainty in the magnitudeof the shape parameter feeds into the particle production calculation, we will consider a fiducial range ofvalues for α c and β c . Estimating / (cid:46) α c , β c (cid:46) and cos θ ab ≈ we find . (cid:46) S (cusp) (cid:46) . Thedimensionless parameters ψ and Θ , given by Eqs. (C.18) and (C.19), are less sensitive to the uncertainty inthe shape parameters. Typically ψ ≈ . and Θ ≈ . .
2. Scalar Integral – Kink
A kink occurs when the derivative of one of the functions b µ ( σ + ) or a µ ( σ − ) appearing in I + ( b ; q ) or I − ( a ; q ) has a discontinuity, and the other integral has a saddle point. For the sake of discussionwe suppose that I + contains the saddle point and I − the discontinuity. We calculate I by insertingEqs. (C.20) and (C.25) in Eq. (C.5). From Eq. (C.20) we see that the leading order term in I + is pro-portional to b (cid:48) s and the subleading term is proportional to b (cid:48)(cid:48) s . Upon contracting with I − the leading orderterm is negligible: we have the identity q · I − = 0 [Eq. (C.4)] and the saddle point criterion q = | q | b (cid:48) s fromwhich it follows that b (cid:48) s · I − = − ( q / | q | − I − ≈ − ( m / | q | ) I − , which is negligible (at | q | > m √ mL )compared to the terms that we keep. The calculation described above yields I (kink) ( q ) = − i B + L / | q | / (cid:2) α + ( b (cid:48)(cid:48) s · a (cid:48) + ) − α − ( b (cid:48)(cid:48) s · a (cid:48)− ) (cid:3) (D.13)Here we have used q ≈ | q | since the saddle point condition requires m √ mL < | q | and mL (cid:29) fortypical size loops. For the same reason, the bound on the discontinuity integral, L − < | q | , is subsumed bythe bound on the saddle point integral, m √ mL < | q | . The squared integral becomes (cid:12)(cid:12) I (kink) ( q ) (cid:12)(cid:12) = S (kink) L / | q | / , ψ m √ mL < | q | , θ < Θ ( | q | L ) − / (band) (D.14)where S (kink) ≡ (2 π ) B β s (cid:2) α + (ˆ b (cid:48)(cid:48) s · a (cid:48) + ) − α − (ˆ b (cid:48)(cid:48) s · a (cid:48)− ) (cid:3) and we have used the shape parameters,introduced in Eq. (C.11). The saddle point criterion requires q to be aligned with b (cid:48) s . Consequently, theradiation is emitted into a band (whose orientation is determined by b (cid:48) s ( σ + ) ) of angular width Θ( | q | L ) − / and angular length ∼ π .We can estimate a range of uncertainty for S (kink) as we did in Appendix D 1. Recall that B + was given by Eq. (C.21). Following the convention established in Appendix D 1, we estimate the shapeparameter as / (cid:46) β s (cid:46) . We also take (cid:46) α ± (cid:46) , as per the discussion below Eq. (C.25). Togetherthis lets us estimate . (cid:46) S (kink) (cid:46) The result of Ref. [27] is derived using this “leading order” term, I + ∼ b (cid:48) s .
3. Scalar Integral – Kink Collision
For the case of a kink-kink collision both integrals, I + and I − , have discontinuities and are givenby Eq. (C.25). The scalar integral is evaluated from Eq. (C.5) to be I (k − k) ( q ) = − ω (cid:104) ( b (cid:48) + · a (cid:48) + )( β + α + ) − ( b (cid:48) + · a (cid:48)− )( β + α − ) − ( b (cid:48)− · a (cid:48) + )( β − α + ) + ( b (cid:48)− · a (cid:48)− )( β − α − ) (cid:105) (D.15)where ω = q . The square is (cid:12)(cid:12) I (k − k) (cid:12)(cid:12) = S (k − k) ω , L − < ω . (D.16)We have defined S (k − k) ≡ (cid:2)(cid:80) ± (1 + b (cid:48)± · a (cid:48)± ) β ± α ± (cid:3) where the sum runs over all possible combina-tions of + and − as given by Eq. (D.15). For the case of a discontinuity, the worldsheet integrals, I ± , areinsensitive to the orientation of k (see Sec. C 2) and the corresponding radiation is emitted approximatelyisotropically.Recall that α ± and β ± were given by Eq. (C.25), and following the conventions established inAppendix D 2, we estimate (cid:46) β ± , α ± (cid:46) . This yields the estimate (cid:46) S (k − k) (cid:46) .
4. Tensor Integral – Cusp
If both I ± contain a saddle point, then we evaluate the tensor integral by inserting Eq. (C.20) intoEq. (D.5). After making use of the identities in Eq. (A.8), many of the terms vanish leaving only Π( q ) = B B − L | q | L ) / (cid:104) ( b (cid:48)(cid:48) c · b (cid:48)(cid:48) c )( a (cid:48)(cid:48) c · a (cid:48)(cid:48) c ) − ( a (cid:48)(cid:48) c · b (cid:48)(cid:48) c ) (cid:105) . (D.17)We extract the factors of L from the bracketed quantities by using the parametrization in Eq. (C.11). Doingso gives Π( q ) (cid:12)(cid:12) (cusp) = T (cusp) L / | q | / , ψ m √ mL < | q | , θ < Θ ( | q | L ) − / (cone) (D.18)where T (cusp) ≡ (2 π ) ( B + B − ) α c β c sin θ ab and θ ab is the angle between a (cid:48)(cid:48) c and b (cid:48)(cid:48) c . The angle between q and b (cid:48) c = − a (cid:48) c is bounded above by Θ( | q | L ) − / , and consequently q is oriented within a cone centeredat the cusp.We can estimate the dimensionless coefficient by making the same estimates as in Appendix D 1.Assuming that the shape parameters fall into the range / < α c , β c < and approximating (1 − cos θ ab ) ≈ we obtain . (cid:46) T (cusp) (cid:46) .1
5. Tensor Integral – Kink If I + contains a saddle point and I − contains a discontinuity, then we evaluate the tensor integralby inserting Eqs. (C.20) and (C.25) into Eq. (D.5). Some of the contractions vanish upon using the identitiesin Eq. (A.8). As we discussed in Sec. D 2, the leading order term in I + is negligible because the contraction b (cid:48) s · I − is suppressed by m / | q | (cid:28) . Making these substitutions we are left with Π( q ) = 14 (cid:34) B L ( b (cid:48)(cid:48) s · b (cid:48)(cid:48) s )( | q | L ) / ( − α + α − ( a (cid:48) + · a (cid:48)− ) | q | − | q | B L ( b (cid:48)(cid:48) s · A (cid:48) ) ( | q | L ) / (cid:35) (D.19)where A (cid:48) ≡ α + a (cid:48) + − α − a (cid:48)− . We relate b (cid:48)(cid:48) s to β s using the parametrization in Eq. (C.11). Then Π( q ) (cid:12)(cid:12) (kink) = T (kink) L / | q | / , ψ m √ mL < | q | , θ < Θ ( | q | L ) − / (band) (D.20)where T (kink) ≡ (2 π ) B β s (cid:32) α + α − (cid:16) (1 − a (cid:48) + · a (cid:48)− ) + (ˆ b (cid:48)(cid:48) s · a (cid:48) + )(ˆ b (cid:48)(cid:48) s · a (cid:48)− ) (cid:17) − α b (cid:48)(cid:48) s · a (cid:48) + ) − α − b (cid:48)(cid:48) s · a (cid:48)− ) (cid:33) . (D.21)The momentum k is constrained to fall within a band of angular width Θ( | k | L ) − / .Following the conventions from the previous sections, we estimate / (cid:46) β s (cid:46) and determine B + from Eq. (C.21). We estimate the parenthetical factor as simply | α + α − | and take (cid:46) α ± (cid:46) as before.Then together we find (cid:46) T (kink) (cid:46) .
6. Tensor Integral – Kink Collision
If both I ± possess a discontinuity point, then we evaluate the tensor integral by inserting Eq. (C.25)into Eq. (D.5). This gives Π( q ) = 14( q ) (cid:104) ( B (cid:48) · B (cid:48) )( A (cid:48) · A (cid:48) ) − ( B (cid:48) · A (cid:48) ) (cid:105) (D.22)where B (cid:48) ≡ β + b (cid:48) + − β − b (cid:48)− and A (cid:48) ≡ α + a (cid:48) + − α − a (cid:48)− . This can be written as Π( k ) (cid:12)(cid:12) (k − k) = T (k − k) q ) , L − < q (D.23)where T (k − k) ≡ (cid:2) β + β − α + α − ( b (cid:48) + · b (cid:48)− )( a (cid:48) + · a (cid:48)− ) − [( β + b (cid:48) + − β − b (cid:48)− ) · ( α + a (cid:48) + − α − a (cid:48)− )] (cid:3) , and we haveused that a (cid:48)± and b (cid:48)± are null vectors.2We can estimate T (k − k) following the conventions established in Appendix D 3. We take (cid:46) α ± , β ± (cid:46) and approximate b (cid:48) + · b (cid:48)− ≈ a (cid:48) + · a (cid:48)− ≈ b (cid:48)± · a (cid:48)± ≈ . This allows us to estimate the range . (cid:46) T (k − k) (cid:46) for the dimensionless coefficient. Appendix E: Cusp Boost Factor and UV Sensitivity
It was recognized in Ref. [24] that particle radiation from a cusp will be highly boosted since thecusp tip moves at the speed of light in the rest frame of the loop. (By contrast, the gravitational radiationspectrum is IR sensitive, and the boost factor is not relevant.) At a given point on the string loop, the boostfactor is given by γ boost ( τ, σ ) = 1 (cid:113) ˙ X µ ( τ, σ ) ˙ X µ ( τ, σ ) = (cid:115) − a (cid:48) ( σ − τ ) · b (cid:48) ( σ + τ ) (E.1)where we have used the formulae in Appendix A. Expanding both a µ and b µ as in Eq. (C.7) and usingEq. (A.8) gives γ boost (∆ σ ) = ( L/ ∆ σ ) π (cid:112) α s + β s (E.2)where ∆ σ is the distance from the tip of the cusp. The dimensionless shape parameters, β s and α s , weredefined in Eq. (C.11). The boost factor grows with decreasing ∆ σ as one investigates radiation comingfrom closer and closer to the tip of the cusp. For a ideal string of zero thickness, we can take ∆ σ → and γ boost → ∞ . In reality, the finite thickness string overlaps with itself at the cusp tip, and a segment of stringwith length ∆ σ min ∼ (cid:112) L/M will evaporate into particle radiation [34]. This leads to an upper bound onthe boost factor, γ boost (cid:46) √ M Lπ (cid:112) α s + β s . (E.3)The radiation spectra that we calculate should drop off when the momentum of the radiated particle exceedsthe inverse string thickness. In the rest frame of the radiating string segment this condition is | k | cusp − frame B827 , 256 (2010), arXiv:0903.1118.[8] E. J. Chun, J.-C. Park, and S. Scopel, JHEP , 100 (2011), arXiv:1011.3300.[9] X. Chu, T. Hambye, and M. H. Tytgat, JCAP , 034 (2012), arXiv:1112.0493.[10] S. Baek, P. Ko, and W.-I. Park, (2014), arXiv:1405.3530.[11] T. Basak and T. Mondal, (2014), arXiv:1405.4877.[12] T. Vachaspati, Phys.Rev. D80 , 063502 (2009), arXiv:0902.1764.[13] J. M. Hyde, A. J. Long, and T. Vachaspati, Phys.Rev. D89 , 065031 (2014), arXiv:1312.4573.[14] B. Patt and F. Wilczek, (2006), arXiv:hep-ph/0605188.[15] B. Holdom, Phys.Lett. B166 , 196 (1986).[16] R. Foot and X.-G. He, Phys.Lett. B267 , 509 (1991).[17] J. Jaeckel, (2013), arXiv:1303.1821.[18] G. Belanger, B. Dumont, U. Ellwanger, J. Gunion, and S. Kraml, Phys.Lett. B723 , 340 (2013), arXiv:1302.5694.[19] A. Vilenkin and Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cam-bridge, UK, 1994).[20] P. Peter, Phys.Rev. D46 , 3322 (1992). [21] M. G. Alford and F. Wilczek, Phys.Rev.Lett. , 1071 (1989).[22] M. Srednicki and S. Theisen, Phys.Lett. B189 , 397 (1987).[23] T. Damour and A. Vilenkin, Phys.Rev.Lett. , 2288 (1997), arXiv:gr-qc/9610005.[24] T. Vachaspati, Phys.Rev. D81 , 043531 (2010), arXiv:0911.2655.[25] E. Sabancilar, Phys.Rev. D81 , 123502 (2010), arXiv:0910.5544.[26] A. J. Long and T. Vachaspati, to appear, (2014) (2014).[27] C. Lunardini and E. Sabancilar, Phys.Rev. D86 , 085008 (2012), arXiv:1206.2924.[28] K. Jones-Smith, H. Mathur, and T. Vachaspati, Phys.Rev. D81 , 043503 (2010), arXiv:0911.0682.[29] Y.-Z. Chu, H. Mathur, and T. Vachaspati, Phys.Rev. D82 , 063515 (2010), arXiv:1003.0674.[30] D. A. Steer and T. Vachaspati, Phys.Rev. D83 , 043528 (2011), arXiv:1012.1998.[31] J. M. Quashnock and T. Piran, Phys.Rev. D43 , 3785 (1991).[32] J. M. Quashnock and D. N. Spergel, Phys.Rev. D42 , 2505 (1990).[33] T. Damour and A. Vilenkin, Phys.Rev. D64 , 064008 (2001), arXiv:gr-qc/0104026.[34] K. D. Olum and J. Blanco-Pillado, Phys.Rev.