Comments on real tachyon vacuum solution without square roots
aa r X i v : . [ h e p - t h ] N ov CCNH-UFABC 2017November, 2017
Comments on real tachyon vacuum solution withoutsquare roots
E. Aldo Arroyo ∗ Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABCSanto Andr´e, 09210-170 S˜ao Paulo, SP, Brazil
Abstract
We analyze the consistency of a recently proposed real tachyon vacuum solution withoutsquare roots in open bosonic string field theory. We show that the equation of motioncontracted with the solution itself is satisfied. Additionally, by expanding the solution inthe basis of the curly L and the traditional L eigenstates, we evaluate numerically thevacuum energy and obtain a result in agreement with Sen’s conjecture. ∗ [email protected] ontents L level expansion analysis of the real solution 115 L level expansion analysis of the real solution 156 Summary and discussion 17 In open string field theory [1], we say that a string field Ψ is real if obeys the followingreality condition Ψ ‡ = Ψ , (1.1)where the double dagger denotes a composition of Hermitian and BPZ conjugation intro-duced in Gaberdiel and Zwiebach’s seminal work [2].Analytic tachyon vacuum solutions that satisfy the above reality condition (1.1) existin the literature [3, 4], however they carry some technical complications. For instance,Schnabl’s original solution is real, but has some subtleties, the solution contains a singular,projector-like state known as the phantom term [5].Solutions without the phantom term, known as simple solutions or Erler-Schnabl’stype solutions have been proposed [6, 7, 8, 9, 10], but they often fail to satisfy the realitycondition. By performing a gauge transformation over a non-real simple solution, a realphantom-less solution has been constructed in reference [6]. However, as noted in reference[11], the cost of having this real solution is the introduction of somewhat awkward squareroots.It would be desirable to have a solution that is both real and simple, namely withoutsquare roots and phantom terms. This is precisely the issue that has been studied in arecent paper [11], where the author has presented an alternative prescription to obtain areal solution from a non-real one which does not make use of a similarity transformation.Basically, it has been shown that given a tachyon vacuum solution Υ together with itscorresponding homotopy operator A [12, 13, 14], the string field defined by Φ = Re(Υ) +Im(Υ) A Im(Υ) is a real solution for the tachyon vacuum.2pplying this prescription for the case of the non-real Erler-Schnabl’s tachyon vacuumsolution [6] Φ
Er-Sch = c (1 + K ) Bc
11 +
K , (1.2)the corresponding real solution [11] has been constructedΦ = 14 (cid:16)
11 +
K c + c
11 + K + c B K c + 11 +
K c
11 + K (cid:17) + Q B -exact terms , (1.3)where the Q B -exact terms are given by12 (cid:2) Q B ( Bc ) 11 + K + 11 + K Q B ( Bc ) (cid:3) + 14 11 + K Q B ( Bc ) 11 + K . (1.4)For this real solution the corresponding energy has been computed and shown that thevalue is in agreement with the value predicted by Sen’s conjecture [15, 16].Nevertheless, for the evaluation of the energy, the equation of motion contracted withthe solution itself was simply assumed to be satisfied. In this paper, we compute thecubic term of the action for the real solution (1.3) and discuss the validity of the previousassumption. Additionally, by expanding the solution in the basis of curly L eigenstates,we evaluate the energy numerically and obtain a result in agreement with Sen’s conjecture.Since the numerical evaluation of the energy by means of the curly L level expansionof the solution is not a trivial task, in order to automate the computations of relevantcorrelation functions defined on the sliver frame, we have developed conservation laws.This paper is organized as follows. In section 2, we evaluate the cubic term of theaction for the real solution and test the validity of the equation of motion when contractedwith the solution itself. In section 3, in order to automate the computations involved in thenumerical evaluation of the energy associated with the solution, we developed conservationlaws for operators defined on the sliver frame. In sections 4, and 5, we compute the energyby means of the curly L and the standard Virasoro L level expansion of the solution andafter using Pad´e approximants we show that the numerical results obtained for the energyare in agreement with Sen’s conjecture. In section 6, a summary and further directionsof exploration are given. In reference [11], a new real solution for the tachyon vacuum has been proposed. Thissolution in the
KBc subalgebra [17, 18] takes the formΦ = 14 (cid:16)
11 +
K c + c
11 + K + c B K c + 11 +
K c
11 + K (cid:17) + Q B -exact terms , (2.1)3y evaluating the kinetic term of the action, it has been shown that the energy E (Φ) = 16 tr[Φ Q B Φ] (2.2)associated with the solution (2.1) correctly reduces to a value which is in accordance withSen’s conjecture.However, to derive the above equation (2.2) for the energy, it has been assumed thatthe equation of motion holds when contracted with the solution itself. We know fromexperience with other solutions [9, 18, 19, 20] that this assumption is not a trivial one.In general, a priori there is no justification for assuming the validity oftr[Φ Q B Φ + ΦΦΦ] = 0 (2.3)without an explicit calculation. Therefore the cubic term of the action must be evaluated.The computation of the kinetic term has been already done in reference [11] given asa result tr[Φ Q B Φ] = − π . (2.4)Thus, for equation (2.3) to be valid, we must show thattr[ΦΦΦ] = 3 π . (2.5)To explicitly compute this cubic term, we need to include the Q B -exact terms of thereal solution (2.1). Recall that these terms were not necessary in the evaluation of thekinetic term. The Q B -exact terms in (2.1) are given by12 (cid:2) Q B ( Bc ) 11 + K + 11 + K Q B ( Bc ) (cid:3) + 14 11 + K Q B ( Bc ) 11 + K . (2.6)Inserting the solution (2.1) which includes the Q B -exact terms (2.6) into the cubicinteraction term tr[ΦΦΦ], after a lengthy algebraic manipulations, we arrive totr[ΦΦΦ] = tr h − cKc
11 +
K c K ) − cKc
11 +
K c
11 + K + 116 B K ) cKc K ) cKc + 316 B K ) cKc
11 +
K cKc + 18 B
11 +
K cKc K ) cKc + 18 B
11 +
K cKc
11 +
K cKc − B
11 +
K c
11 +
K cKc
11 +
K c + 116 B
11 +
K c
11 +
K c
11 +
K cKc i . (2.7)4ll the correlators appearing in the evaluation of the cubic term (2.7) can be computedby means of the following basic correlatorstr (cid:2) ce − t K ce − t K ce − t K (cid:3) = ( t + t + t ) sin (cid:16) πt t + t + t (cid:17) sin (cid:16) πt t + t + t (cid:17) sin (cid:16) π ( t + t ) t + t + t (cid:17) π , (2.8)tr (cid:2) Be − t K ce − t K ce − t K ce − t K c (cid:3) = s ( t + t + t ) π sin( πt s ) sin( πt s ) sin( π ( t + t ) s ) − s ( t + t ) π sin( πt s ) sin( π ( t + t ) s ) sin( π ( t + t + t ) s )+ s t π sin( πt s ) sin( π ( t + t ) s ) sin( π ( t + t + t ) s ) , (2.9)where s = t + t + t + t .For instance, employing the correlator (2.8), let us explicitly compute the correlatortr h cKc
11 +
K c K ) i = − Z ∞ dt dt t e − t − t ∂ s tr h ce − sK ce − t K ce − t K i(cid:12)(cid:12)(cid:12) s =0 = − Z ∞ dt dt t e − t − t ( t + t ) sin (cid:16) πt t + t (cid:17) sin (cid:16) πt t + t (cid:17) π . (2.10)To evaluate the above double integral, we perform the change of variables t → uv , t → u − uv , R ∞ dt dt → R ∞ du R dv u , so that from equation (2.10), we obtaintr h cKc
11 +
K c K ) i = Z ∞ du Z dv e − u u ( v −
1) sin ( πv ) π = − π . (2.11)To compute correlators containing the B string field, we proceed in the same manner.As an illustration, let us explicitly evaluate the correlator tr h B K ) cKc K ) cKc i . Theintegral representation of this correlator is given by Z ∞ dt dt t t e − t − t ∂ s ,s tr h Be − t K ce − s K ce − t K ce − s K c i(cid:12)(cid:12)(cid:12) s = s =0 . (2.12)Using the correlator (2.9), from equation (2.12) we obtain Z ∞ dt dt t t e − t − t (cid:16) πt t + t (cid:17) (cid:16) ( t + t ) sin (cid:16) πt t + t (cid:17) − πt cos (cid:16) πt t + t (cid:17)(cid:17) π . (2.13)5erforming the change of variables t → uv , t → u − uv , R ∞ dt dt → R ∞ du R dv u intothe above double integral (2.13), we get Z ∞ du Z dv e − u u (1 − v ) v sin( πv )(sin( πv ) − π ( v −
1) cos( πv )) π = 30 π + 4 π . (2.14)Therefore, we have just shown thattr h B K ) cKc K ) cKc i = 30 π + 4 π . (2.15)In this way, we can calculate all the relevant correlators appearing in the right handside of equation (2.7). Let us list the resultstr h cKc
11 +
K c K ) i = − π , (2.16)tr h cKc
11 +
K c
11 + K i = − π , (2.17)tr h B K ) cKc K ) cKc i = 30 π + 4 π , (2.18)tr h B K ) cKc
11 +
K cKc i = 3 π , (2.19)tr h B
11 +
K cKc K ) cKc i = 6 π , (2.20)tr h B
11 +
K cKc
11 +
K cKc i = 3 π , (2.21)tr h B
11 +
K c
11 +
K cKc
11 +
K c i = 15 π − π , (2.22)tr h B
11 +
K c
11 +
K c
11 +
K cKc i = − π − π . (2.23)Employing these results (2.16)-(2.23) into equation (2.7) and adding up all terms, weobtain the value for the cubic term tr[ΦΦΦ] = 3 π . (2.24)Since we have explicitly shown that the equation of motion is satisfied when contractedwith the solution itself, i.e. tr[Φ Q B Φ] + tr[ΦΦΦ] = 0, it is guaranteed that the energyassociated with the solution (2.1) is directly proportional to the kinetic term E (Φ) = −S [Φ] = 12 tr[Φ Q B Φ] + 13 tr[ΦΦΦ] = 16 tr[Φ Q B Φ] . (2.25)As a second test of consistency, we would like to analyze the solution from a numericalpoint of view, in particular, we will be interested in the numerical evaluation of the kineticterm by means of the curly L level expansion of the solution.6s we are going to show, when we insert the curly L level expansion of the solutioninto the kinetic term, we are required to evaluate two point vertices for string fieldscontaining the operators ˆ L , ˆ B and ˜ c p . These two point vertices can be evaluated bymeans of the so-called conservation laws which will be studied in the next section. The operators employed in the basis of curly L eigenstates are given in terms of the basicoperators ˆ L , ˆ B and ˜ c p . These operators are related to the worldsheet energy momentumtensor T ( z ), the b ( z ) and c ( z ) ghosts fields respectively. We are going to derive theconservation law for the ˆ L operatorˆ L = I dz πi (1 + z )(arctan z + arccot z ) T ( z ) . (3.1)Using the conformal map ˜ z = π arctan z , we can write the expression of the ˆ L operatorin the sliver frame ˆ L = I d ˜ z πi ε (Re˜ z ) ˜ T (˜ z ) , (3.2)where ε ( x ) is the step function equal to ± φ i defined on the sliver frame, the two functions f and f whichappear in the definition of the two point vertex (cid:10) f ◦ φ (0) f ◦ φ (0) (cid:11) are given by f (˜ z ) = tan (cid:0) π z ) (cid:1) , (3.3) f (˜ z ) = tan (cid:0) π z (cid:1) . (3.4)We need conservation laws such that the operator ˆ L acting on the two point vertex,which we denote as (cid:10) V (cid:12)(cid:12) , can be expressed in terms of non-negative Virasoro modes definedon the sliver frame (cid:10) V (cid:12)(cid:12) ˆ L (2) = (cid:10) V (cid:12)(cid:12)(cid:16) X n ≥ a n L (1) n + X n ≥ b n L (2) n (cid:17) , (3.5)where a n and b n are coefficients that will be determined below. We are going to use the following notation O ( i ) to refer an operator O defined around the i -thpuncture.
7o derive a conservation law of the form (3.5), we need a vector field which behavesas v (2) (˜ z ) ∼ ε (Re˜ z ) + O (˜ z ) around puncture 2, and has the following behavior in theother puncture, v (1) (˜ z ) ∼ O (˜ z ). A vector field which does this job is given by v ( z ) = (1 + z )arccot z. (3.6)The expression of the conservation law for Virasoro modes defined on the sliver frameis given by (cid:10) V (cid:12)(cid:12) X j =1 I C j πi v ( j ) (˜ z j ) ˜ T (˜ z j ) d ˜ z j = 0 , (3.7)where v ( j ) (˜ z j ) = ( ∂ ˜ z j f j (˜ z j )) − v ( f j (˜ z j )), and C j is a closed contour which encircles the j -puncture.Using equations (3.3), (3.4) and (3.6) into the definition v ( j ) (˜ z j ) = ( ∂ ˜ z j f j (˜ z j )) − v ( f j (˜ z j ))of the vector fields v (1) (˜ z ) and v (2) (˜ z ), we find that v (1) (˜ z ) = − ˜ z (3.8) v (2) (˜ z ) = ε (Re˜ z ) − ˜ z . (3.9)Due to the presence of the step function we see that the vector field v (2) (˜ z ) is discontinuousaround puncture 2, since we are interested in the conservation law of the operator definedin equation (3.2), this kind of discontinuity is what we want. Using (3.7) and noting thatintegration amounts to the replacement v ( i ) n ˜ z ni → v ( i ) n L ( i ) n − , we can immediately write theconservation law (cid:10) V (cid:12)(cid:12)(cid:16) − L (1)0 + ˆ L (2) − L (2)0 (cid:17) = 0 . (3.10)We can write this conservation law (3.10) in the standard form as given in equation (3.5) (cid:10) V (cid:12)(cid:12) ˆ L (2) = (cid:10) V (cid:12)(cid:12)(cid:16) L (1)0 + L (2)0 (cid:17) . (3.11)By the symmetry property of the two vertex, the same identity (3.11) holds after replacing(1) → (2) (cid:10) V (cid:12)(cid:12) ˆ L (1) = (cid:10) V (cid:12)(cid:12)(cid:16) L (2)0 + L (1)0 (cid:17) . (3.12)Regarding the conservation law for the ˆ B operator, since the b ghost is a conformalfield of dimension two, the conservation laws for operators involving this field are identical This formula can be derived using the general prescription for conservation laws shown in references[21, 22].
8o those for the Virasoro operators (cid:10) V (cid:12)(cid:12) ˆ B (2) = (cid:10) V (cid:12)(cid:12)(cid:16) B (1)0 + B (2)0 (cid:17) , (3.13) (cid:10) V (cid:12)(cid:12) ˆ B (1) = (cid:10) V (cid:12)(cid:12)(cid:16) B (2)0 + B (1)0 (cid:17) . (3.14)Employing these conservation laws for the operators ˆ L and ˆ B , together with the com-mutator and anti-commutator relations[ L ( i )0 , ˆ L ( j ) ] = δ ij ˆ L ( j ) , [ L ( i )0 , ˆ B ( j ) ] = δ ij ˆ B ( j ) , [ L ( i )0 , ˜ c ( j ) p ] = − δ ij p c ( j ) p , (3.15)[ B ( i )0 , ˆ L ( j ) ] = δ ij ˆ B ( j ) , {B ( i )0 , ˆ B ( j ) } = 0 , {B ( i )0 , ˜ c ( j ) p } = δ ij δ ,p , (3.16)we can show that all two point correlation functions involving string fields constructedout of the operators ˆ L , ˆ B and ˜ c p can be reduced to the evaluation of the following basiccorrelators (cid:10) V (cid:12)(cid:12) ˜ c (2) p ˜ c (2) p ˜ c (2) p (cid:11) = (cid:10) V (cid:12)(cid:12) ˜ c (1) p ˜ c (1) p ˜ c (1) p (cid:11) = I dx dx dx (2 πi ) x p − x p − x p − (cid:10) c ( x ) c ( x ) c ( x ) (cid:11) C , (3.17) (cid:10) V (cid:12)(cid:12) ˜ c (1) p ˜ c (2) p ˜ c (2) p (cid:11) = (cid:10) V (cid:12)(cid:12) ˜ c (1) p ˜ c (1) p ˜ c (2) p (cid:11) = I dx dx dx (2 πi ) x p − x p − x p − (cid:10) c ( x + 1) c ( x ) c ( x ) (cid:11) C , (3.18)where the correlator (cid:10) c ( x ) c ( y ) c ( z ) (cid:11) C L in general is given by (cid:10) c ( x ) c ( y ) c ( z ) (cid:11) C L = L π sin (cid:18) π ( x − y ) L (cid:19) sin (cid:18) π ( x − z ) L (cid:19) sin (cid:18) π ( y − z ) L (cid:19) . (3.19)To evaluate explicitly the above correlators (3.17) and (3.18), the following formulaswill be very useful S a,b ≡ I dz πi z a sin( bz ) = − b − a − cos (cid:0) πa (cid:1) Γ( − a ) , (3.20) C a,b ≡ I dz πi z a cos( bz ) = − b − a − sin (cid:0) πa (cid:1) Γ( − a ) . (3.21)For instance, let us compute correlator (3.17). Using (3.19) into equation (3.17), we have (cid:10) V (cid:12)(cid:12) ˜ c (2) p ˜ c (2) p ˜ c (2) p (cid:11) = (cid:10) V (cid:12)(cid:12) ˜ c (1) p ˜ c (1) p ˜ c (1) p (cid:11) = I dx dx dx (2 πi ) x p − x p − x p − (cid:10) c ( x ) c ( x ) c ( x ) (cid:11) C == 2 π I dx dx dx (2 πi ) x p − x p − x p − h sin ( πx ) cos ( πx ) − sin ( πx ) cos ( πx )+ sin ( πx ) cos ( πx ) − sin ( πx ) cos ( πx )+ sin ( πx ) cos ( πx ) − sin ( πx ) cos ( πx ) i . (3.22)9t is clear that the above equation (3.22) can be written in terms of the functions (3.20)and (3.21), so that we arrive to an explicit expression for the correlator (3.17) (cid:10) V (cid:12)(cid:12) ˜ c (2) p ˜ c (2) p ˜ c (2) p (cid:11) = (cid:10) V (cid:12)(cid:12) ˜ c (1) p ˜ c (1) p ˜ c (1) p (cid:11) = 2 π h δ ,p S p − ,π C p − ,π − δ ,p C p − ,π S p − ,π − δ ,p C p − ,π S p − ,π + δ ,p C p − ,π S p − ,π + δ ,p C p − ,π S p − ,π − δ ,p C p − ,π S p − ,π i . (3.23)In the same way, we can also derive the explicit expression for the correlator (3.18) (cid:10) V (cid:12)(cid:12) ˜ c (1) p ˜ c (2) p ˜ c (2) p (cid:11) = (cid:10) V (cid:12)(cid:12) ˜ c (1) p ˜ c (1) p ˜ c (2) p (cid:11) = 2 π h δ ,p S p − ,π C p − ,π − δ ,p C p − ,π S p − ,π − δ ,p C p − ,π S p − ,π + δ ,p C p − ,π S p − ,π + δ ,p C p − ,π S p − ,π − δ ,p C p − ,π S p − ,π i . (3.24)To evaluate the kinetic term tr[Φ Q B Φ] for a string field Φ expanded in the basis ofcurly L eigenstates, it will be convenient to write the kinetic term in the language of atwo point vertex tr[Φ Q B Φ] = (cid:10) V (cid:12)(cid:12) Φ (1) Q B Φ (2) (cid:11) . (3.25)Note that in addition to the conservation laws, we will be required to know the action ofthe BRST charge Q B on the operators ˆ L , ˆ B and ˜ c p [ Q B , ˆ L ( j ) ] = 0 , { Q B , ˆ B ( j ) } = ˆ L ( j ) , { Q B , ˜ c ( j ) p } = ∞ X k = −∞ (1 − k )˜ c ( j ) p − k ˜ c ( j ) k . (3.26)As an illustration of the use of conservation laws, we are going to compute a particularcorrelator involving the operators ˆ B and ˆ L . We choose, as an example, the following stringfields φ = ˆ B ˆ L ˜ c ˜ c | i , ψ = ˜ c | i . (3.27)Using these string fields, let us evaluate the correlatortr[ φQ B ψ ] = (cid:10) V (cid:12)(cid:12) φ (1) Q B ψ (2) (cid:11) . (3.28)Inserting equation (3.27) into equation (3.28) and using (3.26), we obtaintr[ φQ B ψ ] = − (cid:10) V (cid:12)(cid:12) ˆ B (1) ˆ L (1) ˜ c (1)0 ˜ c (1)1 ˜ c (2)0 ˜ c (2)1 (cid:11) . (3.29)Using the conservation law (3.14) and the anti-commutator relations (3.16), from equation(3.29) we gettr[ φQ B ψ ] = − (cid:10) V (cid:12)(cid:12) ˆ L (1) ˜ c (1)0 ˜ c (1)1 ˜ c (2)1 (cid:11) − (cid:10) V (cid:12)(cid:12) ˆ L (1) ˜ c (1)1 ˜ c (2)0 ˜ c (2)1 (cid:11) − (cid:10) V (cid:12)(cid:12) ˆ B (1) ˜ c (1)0 ˜ c (1)1 ˜ c (2)0 ˜ c (2)1 (cid:11) . (3.30)10mploying the conservation laws (3.12), (3.14) and the commutator and anti-commutatorrelations (3.15), (3.16), from equation (3.30) we arrive totr[ φQ B ψ ] = (cid:10) V (cid:12)(cid:12) ˜ c (1)0 ˜ c (1)1 ˜ c (2)1 (cid:11) + (cid:10) V (cid:12)(cid:12) ˜ c (1)1 ˜ c (2)0 ˜ c (2)1 (cid:11) = 2 (cid:10) V (cid:12)(cid:12) ˜ c (1)1 ˜ c (2)0 ˜ c (2)1 (cid:11) = 2 (cid:16) π (cid:17) = 8 π , (3.31)where we have used equation (3.24). These kind of computations can be automated in acomputer. Next, we are going to apply the results shown in this section to evaluate thekinetic term by means of the curly L level expansion of the real solution (2.1). L level expansion analysis of the real solu-tion Since the kinetic term does not depend on the Q B -exact terms, we are going to consideronly the first term of Φ given in equation (2.1). Let us define this term asˆΦ = 14 (cid:16)
11 +
K c + c
11 + K + c B K c + 11 +
K c
11 + K (cid:17) . (4.1)Using the integral representation of 1 / (1 + K )11 + K = Z ∞ dt e − t (1+ K ) = Z ∞ dt e − t Ω t , (4.2)we can write (4.1) asˆΦ = 14 h Z ∞ dt e − t (cid:16) Ω t c + c Ω t + c Ω t Bc (cid:17) + Z ∞ dsdt e − s − t Ω s c Ω t i . (4.3)By writing the basic string fields K , B in terms of the operators ˆ L , ˆ B , and using themodes ˜ c p of the ghost field c ( z ) defined in the ˜ z -conformal frame ˜ z = π arctan z , we canshow thatΩ t c Ω t Bc Ω t = ∞ X n =0 1 X p = −∞ β n n ! ( x − p + y − p ) ˆ L n ˜ c p | i + ∞ X n =0 1 X p = −∞ X q = −∞ β n n ! ( x − p y − q − x − q y − p ) ˆ B ˆ L n ˜ c p ˜ c q | i , (4.4)where β = 12 −
12 ( t + t + t ) , x = 12 ( t − t − t ) , y = 12 ( t + t − t ) . (4.5)11mploying equation (4.4), it is possible to derive the curly L level expansion ofthe string field defined in equation (4.3). As a pedagogical illustration, let us explicitlycompute the curly L level expansion of the last term appearing on the right hand sideof equation (4.3) Z ∞ dsdt e − s − t Ω s c Ω t = ∞ X n =0 1 X p = −∞ Z ∞ dsdt e − s − t β n n ! ( x − p + y − p ) ˆ L n ˜ c p | i , (4.6)where in this case β = 12 −
12 ( s + t ) , x = y = 12 ( t − s ) . (4.7)As we can see from equations (4.6) and (4.7), we are required to evaluate the followingdouble integral Z ∞ dsdt e − s − t β n n ! x − p = Z ∞ dsdt e − s − t − n + p − ( − s − t + 1) n ( t − s ) − p n ! . (4.8)Performing the change of variables s → uv , t → u − uv , R ∞ dsdt → R ∞ du R dv u intothe above integral (4.8), we obtain Z ∞ dsdt e − s − t β n n ! x − p = Z ∞ du Z dv e − u − n + p − (1 − u ) n u − p (1 − v ) − p n != (( − p −
1) 2 − n + p − ( p − n ! Z ∞ du e − u (1 − u ) n u − p = (( − p −
1) 2 − n + p − ( p − n ! F ( n, − p ) , (4.9)where we have defined F ( M, N ) = Z ∞ du e − u (1 − u ) M u N = M X k =0 ( − M − k (cid:18) Mk (cid:19) ( M + N − k )! (4.10)Proceeding in the same way, we can also calculate the curly L level expansion of thefirst terms appearing on the right hand side of equation (4.3). Adding up all the results,we show that the string field (4.1) has the following curly L level expansionˆΦ = ∞ X n =0 1 X p = −∞ f n,p ˆ L n ˜ c p | i + ∞ X n =0 1 X p = −∞ X q = −∞ f n,p,q ˆ B ˆ L n ˜ c p ˜ c q | i , (4.11)where the coefficients f n,p and f n,p,q are given by f n,p = (1 − ( − p ) 2 − n + p − (cid:16) F ( n, − p ) + − p F ( n, − p ) (cid:17) n ! , (4.12) f n,p,q = (( − q − ( − p ) 2 − n + p + q − F ( n, − p − q ) n ! . (4.13)12o compute the kinetic term, we start by replacing the string field ˆΦ with z L ˆΦ, so thatstates in the curly L level expansion will acquire different integer powers of z at differentlevels. As we are going to see, the parameter z is needed because we need to express thekinetic term as a formal power series expansion if we want to use Pad´e approximants.After doing our calculations, we will simply set z = 1.Let us start with the evaluation of the kinetic term as a formal power series expansionin z . By inserting the expansion (4.11) of the string field ˆΦ into the kinetic term, andusing the conservation laws studied in section 3 to evaluate the corresponding two pointvertices, we obtaintr[ z L ˆΦ Q B (cid:0) z L ˆΦ (cid:1) ] = − π z + (cid:0) − π (cid:1) − z + (cid:0) − π (cid:1) z + (cid:0) −
72 + 19 π (cid:1) z + (cid:0) − π π (cid:1) z + (cid:0) −
36 + 279 π − π (cid:1) z + (cid:0) − π π − π (cid:1) z + · · · (4.14)Considering terms up to order z , and setting z = 1, from equation (4.14) we get 3328%of the expected result (2.4). In principle, we can compute the curly L level expansion ofthe kinetic term up to any desired order, however as we increase the order, the involvedtasks demand a lot of computing time. We have determined the series (4.14) up to order z , and setting z = 1, we obtain about 1 . × % of the expected result. As we cansee, if we naively set z = 1 and sum the series, we are left with a non-convergent result.Recall that in numerical curly L level truncation computations, a regularization tech-nique based on Pad´e approximants provides desired results for gauge invariant quantitieslike the energy [6, 20, 23, 24]. Let us see if after applying Pad´e approximants, we canrecover the expected result.To start with Pad´e approximants, first let us define the normalized value of the kineticterm as follows ˆ E ( z ) ≡ π z z L ˆΦ Q B (cid:0) z L ˆΦ (cid:1) ] . (4.15)Since the series for the kinetic term (4.14) is known up to order z , we can write theseries for ˆ E ( z ) up to order z , and after considering a numerical value for π , we obtainˆ E ( z ) = X E k z k = − . . z − . z − . z + 65 . z − . z + 1445 . z − . z − . z + 218120 .z − . × z + 2 . × z − . × z + 1 . × z − . × z + 1 . × z + 9 . × z − . × z + 3 . × z − . × z . (4.16)13n general, to construct a Pad´e approximant of order P nn ( z ) for the normalized value ofthe kinetic term (4.15), we need to truncate the series (4.16) up to order z n .As an illustration, let us compute the normalized value of the kinetic term using aPad´e approximant of order P ( z ). First, we express ˆ E ( z ) as the rational function P ( z )ˆ E ( z ) = P ( z ) = a + a z + a z b z + b z . (4.17)Expanding the right hand side of (4.17) around z = 0 up to order z and equating thecoefficients of z , z , z , z , z with the expansion (4.16), we get a system of algebraicequations for the unknown coefficients a , a , a , b , and b . Solving those equations weget a = − . , a = − . , a = − . , b = 1 . , b = 4 . . (4.18)Replacing the value of these coefficients inside the definition of P ( z ) (4.17), and evalu-ating this at z = 1, we get the following value P ( z = 1) = − . . (4.19)The results of our calculations are summarized in table 4.1. As we can see, the valueof ˆ E ( z ) at z = 1 by means of Pad´e approximants confirms the expected analytical resultˆ E (1) = π tr[ ˆΦ Q B ˆΦ] → −
1. Although the convergence to the expected answer getsirregular at n = 4, by considering higher level contributions, we will eventually reach tothe right value.Using an alternative resummation technique, we would like to confirm the expectedanswer for the normalized value of the kinetic term. We have used a second methodwhich is based on a combination of Pad´e and Borel resummation. We replace the Boreltransform of ˆ E ( z ), which is defined as ˆ E ( z ) Borel = P E k z k /k !, by its Pad´e approximant P nn ( z ) Borel and then evaluate the integral e P nn ( z ) = Z ∞ dt e − t P nn ( zt ) Borel (4.20)at z = 1. In the third column of table 4.1, we list the results obtained for ˆ E (1) by meansof Pad´e-Borel approximations. Note that starting at the value of n = 4, Pad´e-Borel doesa little better than Pad´e. 14able 4.1: The Pad´e and Pad´e-Borel approximation for the normalized value of the kineticterm ˆ E ( z ) = π z tr[ z L ˆΦ Q B (cid:0) z L ˆΦ (cid:1) ] evaluated at z = 1. The second column showsthe P nn Pad´e approximation. The third column shows the corresponding e P nn Pad´e-Borelapproximation. In the last column, P n represents a trivial approximation, a naivelysummed series. P nn e P nn P n n = 0 − . − . − . n = 2 − . − . − . n = 4 − . − . − . n = 6 − . − . . × n = 8 − . − . . × n = 10 − . − . − . × L level expansion analysis of the real solution To expand the string field (4.3) in the Virasoro basis of L eigenstates, we are going touse the following formulas e − t K ce − t K Bce − t K = r cos (cid:0) πxr (cid:1) (cid:0) π ( r − y ) − r sin (cid:0) πyr (cid:1)(cid:1) π e U r c (cid:0) πxr (cid:1) r ! | i + r cos (cid:0) πyr (cid:1) (cid:0) π ( r + 2 x ) + r sin (cid:0) πxr (cid:1)(cid:1) π e U r c (cid:0) πyr (cid:1) r ! | i + ∞ X k =1 ( − k +1 k − (cid:0) r (cid:1) k − cos (cid:0) πxr (cid:1) cos (cid:0) πyr (cid:1) (4 k − π e U r b − k c (cid:18) r tan (cid:16) πxr (cid:17)(cid:19) c (cid:18) r tan (cid:16) πyr (cid:17)(cid:19) | i , (5.1) r = t + t + t + 1 , x = 12 ( t − t − t ) , y = 12 ( t + t − t ) , (5.2)where the operator e U r is defined as e U r ≡ · · · e u ,r L − e u ,r L − e u ,r L − e u ,r L − e u ,r L − . (5.3)To find the coefficients u n,r appearing in the exponentials, we use r r arctan z ) = lim N →∞ (cid:2) f ,u ,r ◦ f ,u ,r ◦ f ,u ,r ◦ f ,u ,r ◦ f ,u ,r ◦ · · · ◦ f N,u
N,r ( z ) (cid:3) = lim N →∞ (cid:2) f ,u ,r ( f ,u ,r ( f ,u ,r ( f ,u ,r ( f ,u ,r ( · · · ( f N,u
N,r ( z )) . . . ))))) (cid:3) , (5.4)where the function f n,u n,r ( z ) is given by f n,u n,r ( z ) = z (1 − u n,r nz n ) /n . (5.5)15mploying the set of equations (5.1) − (5.3) for the string field (4.3), we obtainˆΦ = Z ∞ dt e − t r sin (cid:0) π r (cid:1) (cid:0) πr − r sin (cid:0) πr (cid:1) + π (cid:1) π e U r (cid:16) c (cid:0) − (cid:0) πt r (cid:1) r (cid:1) + c (cid:0) (cid:0) πt r (cid:1) r (cid:1)(cid:17) + Z ∞ dt ∞ X k =1 e − t ( − k +1 k − (cid:0) r (cid:1) k − sin (cid:0) π r (cid:1) π (4 k − e U r b − k c (cid:0) − (cid:0) πt r (cid:1) r (cid:1) c (cid:0) (cid:0) πt r (cid:1) r (cid:1) + Z ∞ ds Z ∞ dt e − s − t (1 + s + t ) cos (cid:16) π ( t − s )2(1+ s + t ) (cid:17) π e U s + t c (cid:16) π ( t − s )2(1+ s + t ) (cid:17) s + t , (5.6)where r = 1 + t .By writing the c ghost in terms of its modes c ( z ) = P m c m /z m − and employingequations (5.3) and (5.6), the string field ˆΦ can be readily expanded and the individualcoefficients can be numerically integrated. For instance, let us write the expansion of ˆΦup to level fourth statesˆΦ =0 . c | i + 0 . c − | i − . L − c − | i − . b − c c | i− . L − c | i + 0 . L − L − c | i + 0 . c − | i− . b − c − c | i − . L − c − | i + 0 . b − c − c | i + 0 . b − c c | i − . L − b − c c | i + · · · . (5.7)As in the case of the curly L level expansion analysis, to evaluate the normalizedvalue of the vacuum energy, first we perform the replacement ˆΦ → z L ˆΦ and then usingthe resulting string field z L ˆΦ, we define, the analogue of equation (4.15)˜ E ( z ) = π z z L ˆΦ Q B (cid:0) z L ˆΦ (cid:1) ] . (5.8)The normalized value of the vacuum energy is obtained just by setting z = 1. Since thekinetic term is diagonal in L eigenstates, the coefficients of the energy (5.8) at order z L are exactly the contributions from fields at level L . We have expanded the string fieldˆΦ given in equation (5.6) up to level twelfth states, and hence the series of ˜ E ( z ) can bedetermined up to the order z ˜ E ( z ) = − . − . z − . z + 0 . z − . z + 0 . z − . z . (5.9)If we naively evaluate the truncated vacuum energy (5.9), i.e., setting z = 1 in theseries before using Pad´e or Pad´e-Borel approximations, we obtain a non-convergent result.Note that the series (5.9) is less divergent than the series (4.16) that has been obtainedin the case of the curly L level expansion analysis of the energy.Let us re-sum the divergent series (5.9). To obtain the Pad´e or Pad´e-Borel approxi-mation of order P nn for the energy, we will need to know the series expansion of ˜ E ( z ) upto the order z n . The results of these numerical calculations are summarized in table 5.1.16able 5.1: The Pad´e and Pad´e-Borel approximation for the normalized value of the vac-uum energy ˜ E ( z ) = π z tr[ z L ˆΦ Q B (cid:0) z L ˆΦ (cid:1) ] evaluated at z = 1. The second column showsthe P nn Pad´e approximation. The third column shows the corresponding e P nn Pad´e-Borelapproximation. In the last column, P n represents a trivial approximation, a naivelysummed series. P nn e P nn P n n = 0 − . − . − . n = 4 − . − . − . n = 8 − . − . − . n = 12 − . − . − . We have analyzed the validity of the recently proposed real tachyon vacuum solution [11],in open bosonic string field theory. We have found that the solution solves in a non trivialway the equation of motion when contracted with itself. Let us point out that a similartest of consistency was performed by Okawa [18], Fuchs, Kroyter [19] and Arroyo [20] forthe case of the original Schnabl’s solution [3].As a second test of consistency, we have analyzed the solution from a numerical pointof view. Using either the curly L , or the Virasoro L level expansion of the solution, wehave found that the expression representing the energy is given in terms of a divergentseries, which nevertheless can be re-summed, either by means of Pad´e technique or acombination of Pad´e-Borel resummation to bring the expected result in agreement withSen’s conjecture.It would be interesting to analyze other real solutions. For instance, the tachyonvacuum solution corresponding to the regularized identity based solution [8]. The realversion of this solution, obtained by means of a similarity transformation, contains squareroots and consequently the analytical and numerical computations of the energy becomecumbersome [9, 23]. Employing the prescription studied in reference [11], it should bepossible to find an alternative real version for this regularized identity based solution.Finally, regarding to the modified cubic superstring field theory [25] and Berkovits non-polynomial open superstring field theory [26], since these theories are based on Witten’sassociative star product, their mathematical setup shares the same algebraic structure ofthe open bosonic string field theory, and thus the prescription developed in reference [11]and the results shown in this paper should be extended to construct and study new realsolutions in the superstring context like the ones discussed in references [24, 27, 28, 29,30, 31, 32]. 17 cknowledgements I would like to thank Ted Erler and Max Jokel for useful discussions.
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