Symmetric formulation of neutrino oscillations in matter and its intrinsic connection to renormalization-group equations
aa r X i v : . [ h e p - ph ] F e b Symmetric formulation of neutrino oscillations in matter andits intrinsic connection to renormalization-group equations
Shun Zhou ∗ Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, ChinaCenter for High Energy Physics, Peking University, Beijing 100871, China
Abstract
In this article, we point out that the effective Hamiltonian for neutrino oscillationsin matter is invariant under the transformation of the mixing angle θ → θ − π/ m ↔ m , if the standard parametrizationof lepton flavor mixing matrix is adopted. To maintain this symmetry in perturbativecalculations, we present a symmetric formulation of the effective Hamiltonian by in-troducing an η -gauge neutrino mass-squared difference ∆ ∗ ≡ η ∆ + (1 − η )∆ for0 ≤ η ≤
1, where ∆ ji ≡ m j − m i for ji = 21 , ,
32, and show that only η = 1 / η = cos θ or η = sin θ is allowed. Furthermore, we prove that η = cos θ isthe best choice to derive more accurate and compact neutrino oscillation probabilities,by implementing the approach of renromalization-group equations. The validity ofthis approach becomes transparent when an analogy is made between the parameter η herein and the renormalization scale µ in relativistic quantum field theories. PACS number(s): 14.60.Pq, 25.30.Pt ∗ E-mail: [email protected] ntroduction — Neutrino oscillation experiments in the last few decades have providedus with compelling evidence for tiny neutrino masses and significant lepton flavor mixing.This great achievement in elementary particle physics has been recognized by the NobelPrize in Physics in 2015 [1, 2]. In the framework of three neutrino flavors, lepton flavormixing can be described by a 3 × U , i.e., the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [3, 4], which is usually parametrized in terms of three mixing angles { θ , θ , θ } and one CP-violating phase δ . Adopting the standard parametrization advo-cated by the Particle Data Group [5], we have U = R ( θ ) · R ( θ , δ ) · R ( θ ) ≡ c s − s c c s e − i δ − s e i δ c c s − s c
00 0 1 , (1)where c ij ≡ cos θ ij and s ij ≡ sin θ ij have been defined for ij = 12 , , R ( θ ij ) denotes arotation matrix in the i - j plane with a rotation angle θ ij , and R ( θ , δ ) = U δ R ( θ ) U † δ with U δ ≡ diag { , , e i δ } . At present, three mixing angles θ ≈ ◦ , θ ≈ ◦ and θ ≈ ◦ ,together with two neutrino mass-squared differences ∆ ≡ m − m ≈ . × − eV and | ∆ | ≡ | m − m | ≈ . × − eV , have been well determined from neutrino oscillationexperiments [5]. The primary goals of future experiments are to pin down neutrino massordering, i.e., the sign of ∆ , and to probe the leptonic CP-violating phase δ .To achieve these goals, the ongoing and forthcoming oscillation experiments are designedfor medium- or long-baseline lengths, and neutrino beams are actually propagating throughthe Earth. In this case, the impact of a coherent forward scattering of neutrinos with back-ground electrons can be taken into account by an effective matter potential V = √ G F N e ,where G F = 1 . × − GeV − is the Fermi constant and N e stands for the net electronnumber density. It is well known that the matter potential can dramatically modify neu-trino flavor conversions [6, 7]. For antineutrinos, the matter potential will change to a minussign. Considering a neutrino beam of energy E travelling in matter, we can write down theeffective Hamiltonian for neutrino flavor oscillations [8, 9, 10] e H eff = 12 E U m m
00 0 m U † + A ≡ e Ω ν E , (2)with A ≡ EV and e Ω ν being defined as the square of the effective neutrino mass matrix inmatter. As usual, one can diagonalize the effective Hamiltonian by the corresponding PMNSmatrix e U in matter, namely, e Ω ν = e U e m e m
00 0 e m e U † , (3)where e m i for i = 1 , , e U can be parametrized in termsof effective mixing parameters { e θ , e θ , e θ } and e δ in the same way as U in Eq. (1).2ith the help of three effective neutrino masses e m i and the flavor mixing matrix e U ,it is straightforward to calculate neutrino oscillation probabilities for a constant matterdensity [11]. Moreover, based on the structure of e H eff and its relation to the Hamiltonian invacuum, one can derive the Naumov relation e J e ∆ e ∆ e ∆ = J ∆ ∆ ∆ [12, 13, 14, 15],where e J and J are respectively the Jarlskog invariants in matter and in vacuum [16], andalso obtain the Toshev relation sin 2 e θ sin e δ = sin 2 θ sin δ [17, 18]. These identities are veryuseful in understanding the relationship between matter-corrected mixing parameters andthe intrinsic ones. Symmetric formulation — In practice, it is necessary to express the oscillation prob-abilities in terms of { θ , θ , θ , δ } and { ∆ , ∆ } , which are the fundamental parametersto be extracted from oscillation experiments. To this end, we can follow a direct diagonal-ization of e H eff or equivalently e Ω ν and calculate the eigenvalues and eigenvectors. Then, thederived exact oscillation probabilities can be expanded in terms of some small parameters.Before doing so, we should first explore the basic properties of the effective Hamiltonian, byrecasting e Ω ν into the following form e Ω ν = c s − s c
00 0 1 m m
00 0 m c − s s c
00 0 1 + A c c s c s s , (4)where the standard parametrization in Eq. (1) is taken and a unitary transformation in theflavor space e Ω ν = [ R † ( θ ) · U † δ · R † ( θ )] · e Ω ν · [ R ( θ ) · U δ · R ( θ )] is performed. The factthat U † δ and U δ commute with R ( θ ) and the diagonal matrix diag { m , m , m } should benoted as well.The transformation in the flavor space by a unitary matrix ˆ U ≡ R ( θ ) · U δ · R ( θ ) doesnot affect the eigenvalues of e Ω ν = ˆ U e Ω ν ˆ U † . Given e Ω ν = U · diag { e m , e m , e m } · U † , one canget the final mixing matrix e U = ˆ U U . From the first part on the right-hand side of Eq. (4),we can identify an intrinsic symmetry under θ → θ − π , m ↔ m , (5)indicating { s , c } → {− c , s } and { sin 2 θ , cos 2 θ } → {− sin 2 θ , − cos 2 θ } for themixing angle, and ∆ → − ∆ for the mass-squared difference. It is easy to verify that theeffective Hamiltonian e H eff is invariant under these transformations. Note that if a differentparametrization of U is assumed, the transformations will be changed to those associatedwith the rightmost rotation matrix in U and the corresponding mass eigenvalues.One may argue that such a symmetry is spurious in the sense of just changing theparameter space from one part to another [19]. But this is not the case. To clearly see thispoint, we follow Ref. [20] and discuss the physical ranges of θ and ∆ . First of all, thereare two different ways to define neutrino mass eigenstates: (A) ν is lighter than ν , i.e.,∆ >
0; (B) ν contains more component of ν e , i.e., | U e | = c > | U e | = s . Then,one can determine the physical ranges of θ and ∆ : θ ∈ [0 , π/
2] and ∆ > θ ∈ [0 , π/
4] and either ∆ > < θ → − θ can3e compensated by redefining the phases of charged-lepton and neutrino fields. Moreover,as proved in Ref. [20], the points ( θ , ∆ ) and ( π/ − θ , ∆ ) in Case (A) are equivalentto ( θ , ∆ ) and ( θ , − ∆ ) in Case (B). Therefore, the transformations in Eq. (5) andthe equivalence between the parameter space in Case (A) and Case (B) can be summarizedvisually in a simple diagram( θ , ∆ ) | ( A ) ✤✤ θ − π/ + ( π/ − θ , ∆ ) | ( A ) K S (cid:11) (cid:19) ( θ , ∆ ) | ( B ) (cid:11) (cid:19) K S k s m ↔ m ✤✤ ( θ , − ∆ ) | ( B ) implying that the whole system should be invariant no matter which definition of neutrinomass eigenstates is taken.For later convenience, we introduce a gauge parameter η ∈ [0 ,
1] and separate an identitymatrix from e Ω ν , namely, e Ω ν = (cid:2) ηm + (1 − η ) m (cid:3) + Ac + ( η − c )∆ ∆ s c As c ∆ s c ( η − s )∆ As c As + ∆ ∗ , (6)where ∆ ∗ = η ∆ + (1 − η )∆ . The definition of ∆ ∗ has been discussed by Parke [21] andhis collaborators [22, 23]. In particular, it has been demonstrated that ∆ c ≡ c ∆ + s ∆ is more advantageous than any other combinations of ∆ and ∆ in description of reactorneutrino experiments [21]. More recently, it has been found in Ref. [24] that ∆ c can beimplemented to greatly simplify the neutrino oscillation probabilities in matter, when thelatter are expanded in terms of the small ratio α c ≡ ∆ / ∆ c ≈ .
03. However, the underlyingreason for this simplification is not well justified in Ref. [24].Now we have a closer look at the new form of e Ω ν in Eq. (6). Since the effective Hamil-tonian possesses an intrinsic symmetry under the transformations θ → θ − π/ m ↔ m (i.e., ∆ → − ∆ ), it should also be respected by the manual separation inEq. (6). Retaining this symmetry in each part, we find only three solutions for η : • mean scheme – η = 1 / m ≡ ∆ ∗ ( η = 1 /
2) = (∆ + ∆ ) /
2. In this scheme, wecan obtain e Ω ν = m + m + ∆ m b A m c − α m c θ / α m s θ / b A m s c α m s θ / α m c θ / b A m s c b A m s + 1 , (7)where b A m ≡ A/ ∆ m and α m ≡ ∆ / ∆ m . This definition of ∆ m has already been usedby the Bari group for a global-fit analysis of neutrino oscillation data [25]. • cosine scheme – η = c and ∆ c ≡ ∆ ∗ ( η = c ) = c ∆ + s ∆ . This choice has beenadopted in a number of works by Parke and others [21, 22, 23, 24]. For this scheme,4e can get e Ω ν = (cid:0) m c + m s (cid:1) + ∆ c b A c c α c s θ / b A c s c α c s θ / α c c θ b A c s c b A c s + 1 , (8)where b A c ≡ A/ ∆ c and α c ≡ ∆ / ∆ c . In the following two sections, we try to explainwhy the series expansions of oscillation probabilities in this scheme give us the mostaccurate and compact results. • sine scheme – η = s and ∆ s ≡ ∆ ∗ ( η = s ) = s ∆ + c ∆ . In this scheme, wecan obtain e Ω ν = (cid:0) m s + m c (cid:1) + ∆ s b A s c − α s c θ α s s θ / b A s s c α s s θ / b A s s c b A s s + 1 , (9)where b A s ≡ A/ ∆ s and α s ≡ ∆ / ∆ s . This definition has also been used for seriesexpansions of neutrino oscillation probabilities that are numerically studied in Ref. [24].Though all the formulas in Eqs. (7), (8) and (9) are equivalent to the original one in Eq. (4),one can observe that each matrix element in e Ω ν in the symmetric formulation respectsthe symmetry indicated in Eq. (5). As a consequence, the parameters α ’s are now alwayscombined with either s θ ≡ sin 2 θ or c θ ≡ cos 2 θ to form an invariant.For comparison, we also explicitly write down e Ω ν in the normal scheme with η = 1, i.e., e Ω ν = m + ∆ b Ac + αs αs θ / b As c αs θ / αc b As c b As + 1 , (10)where α ≡ ∆ / ∆ and b A ≡ A/ ∆ have been defined. It is straightforward to observe therelation ∆ ∗ = ∆ [1 − (1 − η ) α ] and the “renormalization” of two important parameters α ∗ = α − (1 − η ) α , b A ∗ = b A − (1 − η ) α , (11)where the subscripts “ ∗ ” should be replaced by their counterparts in the symmetric schemes.The eigenvalues of e Ω ν can be calculated even without any specific parametrization of thePMNS matrix, and in a way independent of flavor basis [26, 27, 14]. However, here we areinterested in the symmetric form in the standard parametrization of U , namely, e m = (cid:2) ηm + (1 − η ) m (cid:3) + 13 x − p x − y h z + p − z ) i , e m = (cid:2) ηm + (1 − η ) m (cid:3) + 13 x − p x − y h z − p − z ) i , e m = (cid:2) ηm + (1 − η ) m (cid:3) + 13 x + 23 z p x − y , (12)5here x , y and z are given by x = ∆ ∗ h b A ∗ + (2 η − α ∗ i ,y = ∆ ∗ n b A ∗ c + α ∗ h η −
1) + b A ∗ c θ c + b A ∗ (2 η − s ) i + η ( η − α ∗ o ,z = cos
13 arccos 2 x − xy + 27 α ∗ ∆ ∗ h ( η − s ) b A ∗ c + η ( η − b A ∗ s ) α ∗ i x − y ) / . (13)It is worth mentioning that x and y depend on the gauge parameter η , whereas x − y and z actually do not if they are expressed in terms of the original parameters ∆ , α and A . Thedependence on η comes into play when we use ∆ ∗ , α ∗ and b A ∗ and perform series expansionsof the eigenvalues in terms of α ∗ . Series expansions — It has been a longstanding problem in neutrino physics to derivemore accurate and compact formulas for neutrino oscillation probabilities in matter, whichcould help explain the experimental results. One practically useful approach is to expand theoscillation probabilities in terms of some small parameters, e.g., the ratio of two hierarchialneutrino mass-squared differences α ≡ ∆ / ∆ ≈ .
03 and the smallest mixing angle s ≡ sin θ ≈ .
02 in the standard parametrization of U . See, e.g., Refs. [28, 29, 30] for earlydevelopment along this direction, and Ref. [31, 32, 22, 33, 34, 23, 24] for recent progress.For our purpose, it is instructive to concentrate first on two important functions p x − y and z appearing in the mass eigenvalues in Eq. (12). The exact formulas of them can bedirectly computed by using Eq. (13), while their series expansions up to the second order of α ∗ have been given in Ref. [24]. To the first order of α ∗ , one can get z ≈ b A ∗ + 3 b C ∗ b C ′∗ + α ∗ b C ∗ b C ′∗ h b C ∗ (1 − η ) − η − c )(1 − b A ∗ c θ − b C ∗ ) i − α ∗ (1 + b A ∗ + 3 b C ∗ )8 b C ′ ∗ h (1 − η )(1 + b A ∗ ) + 3 b A ∗ c ( η − c ) i , (14)and p x − y ≈ ∆ ∗ (cid:26) b C ′ + α ∗ b C ′ h (1 − η )(1 + b A ∗ ) + 3 b A ∗ c ( η − c ) i(cid:27) , (15)where b C ∗ ≡ [(1 − b A ∗ ) + 4 b A ∗ s ] / and b C ′∗ ≡ ( b C ∗ + b A ∗ c ) / have been introduced. Someinteresting observations are summarized below: • Setting η = 1 / η = c , one can see that all the terms proportional to 1 − η or η − c will disappear, leading to a great simplification of the approximate results inEqs. (14) and (15). If we take another value η = s , both 1 − η and η − c givethe same factor cos 2 θ up to a sign, so those two terms in the square brackets on theright-hand side of Eqs. (14) and (15) can be combined into a single one. In this sense,the choice of η in all three symmetric schemes help derive simpler analytical results.6 One can compute three eigenvalues to the first order of α ∗ with the help of Eqs. (14)and (15). For illustration, we only quote the approximate result for e m from Ref. [24] e m ≈ m − η ∆ + ∆ ∗ " b A ∗ + b C ∗ − ( η − c )(1 − b C ∗ − b A ∗ c θ )2 b C ∗ α ∗ , (16)which can reproduce the same result in Ref. [29] by setting η = 1, namely, e m ≈ m + ∆ " b A + b C s ( b C − b Ac θ )2 b C α . (17)On the other hand, in the cosine scheme with η = c , one can see the first-order termvanishes, and the leading-order contribution reads e m ≈ m + s ∆ + ∆ c b C c + b A c , (18)where b C c = [(1 − b A c ) + 4 b A c s ] / is implied. Therefore, higher-order terms start from O ( α ∗ ) in the cosine scheme. In order to clarify that the leading-order result in Eq. (18)is even more precise than that in Eq. (17), we recall the definitions ∆ c ≡ ∆ (1 − s α )and b A c ≡ b A/ (1 − s α ) and insert them into Eq. (18). Expanding the function b C c tothe second order of α , we arrive at e m ≈ m + ∆ " b A + b C s ( b C − b Ac θ )2 b C α + ∆ " s ( b C − b Ac θ )( b C + 1 − b Ac θ )4 b C α + O ( α ) , (19)which exactly reproduces the first-order result in Eq. (17) and partly incorporates thesecond-order corrections. This can explain why the numerical precision in the cosinescheme is superior to that in the normal scheme, when the oscillation probabilities areexpanded to the same order.In a similar way, one can derive the results for η = 1 / η = s and compare themwith those in Eq. (19). Although the first-order terms are not vanishing in the mean andsine schemes, the final results involving the “renormalized” parameters α m and b A m (or α s and b A s ) can also be regarded as a resummation of higher-order terms of α . Since all threeeigenvalues and oscillation probabilities have been given in Ref. [24] for the general η gauge,it is unnecessary to repeat them here. Renormalization-group equations — Though we have seen that η = c gives riseto the simplest results, as the first-order correction is vanishing, it is not understood why itshould be so. From the symmetry arguments in the previous section, three schemes shouldbe equally powerful in simplifying approximate formulas. In the following, we explain thereason by implementing the renormalztion-group equations (RGEs), which have been widely7pplied in quantum field theories [35, 36] and condense matter physics [37]. In our case, thecentral idea is that the exact mass eigenvalues of e H eff should be independent of the gaugeparameter η . In fact, however, they are computed via perturbative expansions, and thedependence on η actually comes in at any given order of α ∗ .Assuming now η to be an arbitrary positive parameter, which acts like the renormalizationscale µ in relativistic quantum field theories, we shall examine the η -dependence of masseigenvalues e m i . First, as indicated in Eq. (11), the exact dependence of α ∗ and b A ∗ on η isalready known, and can be reflected by the following RGEsd α ∗ d η = − α ∗ , d b A ∗ d η = − b A ∗ α ∗ , (20)where we have used the “renormalized” parameters α ∗ and b A ∗ in the beta functions on theright-hand side of Eq. (20). Notice that these RGEs are the exact results, so we are actuallydealing with an exactly solvable model. Then, it is easy to derive the RGE of b C ∗ from itsdefinition b C ∗ = (1 − b A ∗ ) + 4 b A ∗ s , i.e.,d b C ∗ d η = b A ∗ − c θ b C ∗ d b A ∗ d η = − b A ∗ ( b A ∗ − c θ ) b C ∗ α ∗ . (21)The exact solutions to these RGEs are actually the definitions of α ∗ , b A ∗ and b C ∗ with α ∗ = α , b A ∗ = b A and b C ∗ = b C at η = 1.Second, the RGEs can be used to investigate the η -dependence of the eigenvalues e m i . Wetake e m for an illustrative example, and its approximate formula has been given in Eq. (16).At the leading order of α ∗ , we calculate the derivative of f (0) ( η ) ≡ ( e m − m ) / ∆ , wherethe superscript “(0)” means that the zeroth-order term in e m is included. The final result isdd η f (0) ( η ) = 12 [1 − (1 − η ) α ] " ( b A ∗ + b C ∗ − α ∗ + d b A ∗ d η + d b C ∗ d η . (22)Requiring d f (0) / d η = 0 and making use of the first identity in Eq. (21), one arrives atd b A ∗ d η = − ( b A ∗ + b C ∗ − b C ∗ b A ∗ + b C ∗ − c θ α ∗ , (23)which is different from the exact result of d b A ∗ / d η in Eq. (20). This is reasonable becauseonly the leading-order contribution is taken into account. Moreover, the RGE of α ∗ is notinvolved at the leading order, which is also evident from its exact formula in Eq. (20).Then we go to the first order of α ∗ , and define the function f (1) ( η ) ≡ ( e m − m ) / ∆ ,which now includes both leading- and first-order terms. After a quick calculation, we finddd η f (1) ( η ) = 12 [1 − (1 − η ) α ] × (" ( b A ∗ + b C ∗ − − − b C ∗ − b A ∗ c θ b C ∗ α ∗ − ( η − c )(1 − b C ∗ − b A ∗ c θ ) b C ∗ (cid:18) α ∗ + d α ∗ d η (cid:19) (24)+ d b A ∗ d η η − c b C ∗ c θ α ∗ ! + d b C ∗ d η " η − c )(1 − b A ∗ c θ ) b C ∗ α ∗ . α ∗ , b A ∗ and b C ∗ from Eqs. (20) and (21) into Eq. (24), we obtaina considerably simple resultdd η f (1) ( η ) = 12 [1 − (1 − η ) α ] ( η − c ) b A ∗ s θ α ∗ . (25)This implies that the requirement for d f (1) / d η = 0 at the first order of α ∗ is consistent withthe exact RGEs of α ∗ , b A ∗ and b C ∗ for η = c , resembling the main feature of the exactformula of e m , i.e., d e m / d η = 0, in this case. Therefore, any higher-order contributionsto the beta functions of b A ∗ and α ∗ will either vanish or be proportional to ( η − c ) n with n being a positive integer. For other different values of η , one should derive the RGEs of b A ∗ and α ∗ order by order until the exact results in Eq. (20) are reached. This observationgives a severe constraint on the structure of higher-order terms, and demonstrates that theperturbation results for the choice of η = c are much simpler. It is interesting to applythis approach to other mass eigenvalues and also the oscillation probabilities.A brief comparison between our findings with the existing results in Refs. [21, 22, 23, 24]should be helpful. Although the advantages of η = c in deriving compact and accurateformulas of neutrino oscillation probabilities have been emphasized in those works, it hasnot been observed that the underlying reason may be due to an intrinsic symmetry in theeffective Hamiltonian and the η -dependence of higher-order terms in series expansions canbe studied in a convenient way by implementing the RGE approach. Summary — We have pointed out that the effective Hamiltonian for neutrino oscillationsin matter possesses an intrinsic symmetry under the transformations θ → θ − π/ m ↔ m , if the standard parametrization of the PMNS matrix is adopted. Based onthis symmetry, we suggest an introduction of the η -gauge neutrino mass-squared difference∆ ∗ ≡ η ∆ + (1 − η )∆ and advocate three schemes with η = 1 / η = c and η = s , forwhich such a symmetry is respected at any order of perturbative expansions of α ∗ ≡ ∆ / ∆ ∗ .The expansion in terms of α ∗ in such a symmetric formulation actually incorporates manyhigher-order terms of α . This follows the spirit of resummation.The effective Hamiltonian e H eff can be exactly solved for a constant matter density. In thisexact formulation, the eigenvalues and the corresponding eigenvectors are independent of thegauge parameter η , so are the oscillation probabilities. It becomes important only when wecalculate the physical quantities by using the perturbation theory, i.e., series expansions interms of α ∗ . Therefore, a symmetric formulation does make sense.We have shown that all three symmetric schemes are helpful in simplifying the analyticalresults, and provide a simple proof for η = c as the best choice, following the idea ofrenormalization-group equations. Noticing that α c = α/ (1 − s α ) itself in the cosine schemewith η = c can be expanded in terms of α , and likewise for b A c = b A/ (1 − s α ), we doexpect that the numerical accuracy in this scheme is higher, as s ≈ . α and it is the smallest compared to its counterparts 0 . c ≈ . cknowledgements The author is indebted to Yu-feng Li, Jue Zhang and Jing-yu Zhu for fruitful collaborationand intensive discussions on neutrino oscillations in matter, and to Prof. Zhi-zhong Xing forhelpful comments. This work was supported in part by the National Recruitment Programfor Young Professionals and by the CAS Center for Excellence in Particle Physics (CCEPP).
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