Lorentz Violation, Bhabha Scattering and Finite Temperature

Home , Bhabha scattering

International Centre for Physics Instituto de F´ısica,Universidade de Bras´ılia Ano VII, Fevereiro de 2019 • http://periodicos.unb.br/index.php/e-bfis • eBFIS 7 7241-1(2019)

Lorentz violation, Bhabha scattering and ﬁnite temperature

A. F. Santos∗ Instituto de F´ısica, Universidade Federal de Mato Grosso, 78060-900, Cuiab´a,Mato Grosso, Brazil

In this paper an introduction to Lorentz violation has been done. Corrections due to Lorentz violation for the cross section of the Bhabha scattering has been calculated at zero and ﬁnite temperature. The ﬁnite temperature corrections are calculated using the Thermo Field Dynamics (TFD) formalism.

Keywords: Thermo Field Dynamics (TFD), Bhabha scattering

I. INTRODUCTION zero and finite temperature is calculated. Lorentz symmetry violation is expected to be small at very high ener- 19 The Standard Model (SM) is a theory that describes gies, i.e., Planck energies (∼ 10 GeV). But this is not all the elementary particles and its interactions. Lorentz valid in all cases. It is likely that Lorentz violation ope- and CPT symmetry have been the foundation of the SM. rators with dimension d > 4 will be relevant in searches Although successfully confirmed, the SM is not a funda- involving very high energies [9]. Although Bhabha scat- mental theory since gravity is not included. A fundamen- tering, at high energy in colliders like LEP, are still at tal theory that unifies SM and gravity would emerge at zero temperature, there is certainly no investigation at energies approaching Planck scale (≈ 1019 GeV). Tiny extremely high energy with non-zero temperature. Even violations of Lorentz and CPT symmetries could emerge though these are small numbers still it is important to in models unifying gravity with quantum physics such as calculate the role of temperature in Bhabha Scattering. string theory [1]. This would involve finding new phy- These estimates will give us a reasonable idea of the role sics. For study such violations a new theory has been of SME at finite temperatures. developed that is called the Standard Model Extension There are three different, but equivalent, formalisms (SME). to introduce temperature effects in a quantum field the- The SME [2, 3] contains the Standard Model, Gene- ory. (i) The Matsubara formalism [10] which is based ral Relativity and all possible operators that break Lo- on a substitution of time, t, by a complex time, iτ. (ii) rentz symmetry. In addition, the SME is divided into The closed time path formalism [11] that is a real time two parts: (i) the minimal version which has operators formalism at finite temperature. (iii) The Thermo Field with dimensions d ≤ 4 and preserves conventional quan- Dynamics (TFD) formalism [12–16]. TFD formalism is a tization, hermiticity, gauge invariance, power counting real time finite temperature formalism. Here this forma- renormalizability, and positivity of the energy and (ii) lism is considered. TFD has as basic ingredients: (i) the the non-minimal version which also includes operators doubling of the original Fock space and (ii) the Bogoliu- of higher dimensions. The structure of the SME is one bov transformation. This doubling consists of Fock space ˜ way to investigate the Lorentz violation. Another inte- composed of the original, S and the tilde space S. The ˜ resting way is to modify the interaction vertex between map between the tilde Ai and non-tilde Ai operators is fermions and photons, i.e., a new non-minimal coupling defined by the following tilde (or dual) conjugation rules: term added to the covariant derivative. The non-minimal ∼ ˜ ˜ ˜ ∼ coupling term may be CPT-odd or CPT-even. (AiAj) = AiAj, (Ai) = −ξAi, (1) The main objective of this work is to study corrections † † ∗ (A )∼ = A˜i , (cAi + Aj)∼ = c A˜i + A˜j, for the Bhabha scattering [4] due to Lorentz violation. It i is a process usually used in test of experiments at high with ξ = −1 for bosons and ξ = +1 for fermions. The energy accelerators [5–8]. The cross section for this scat- physical variables are described by non-tilde operators. tering in the context of non-minimal coupling term at The Bogoliubov transformation is a rotation involving these two spaces. As a consequence the propagator is written in two parts: T = 0 and T 6= 0 components. This paper is organized as follows. In section II, the ∗ alesandroferreira@fisica.ufmt.br Lagrangian of the Standard Model Extension is briefly A. F. Santos discussed. The non-minimal Lorentz-violating coupling where is considered. In section III, the Bhabha scattering at zero temperature with Lorentz violation is determined. µ µ µν µν µ µ 1 κµν In Section IV, a brief introduction to the TFD formalism Γ = γ + (c + d γ5)γν + e + if γ5 + g σκν(4), 2 is presented. In section V, the cross section for Bhabha µ µ 1 µν scattering with Lorentz violation at finite temperature is M = m + (a + b γ5)γµ + H σµν . (5) calculated. In section VI, some concluding remarks are 2 presented. The parameters in Γµ are dimensionless while the ones µ in M have dimension of mass. γ , γ5 and σκν denote the II. INTRODUCTION TO THE STANDARD Dirac matrices. The coefficients for Lorentz violation are MODEL EXTENSION aµ, bµ, cµν , dµν , eµ, f µ, gκµν and Hµν . B. The pure-photon sector of the minimal-SME Here a brief introduction to the Lagrangian of the SME is presented. The SME is a field-theory based test fra- The lagrangian of the pure photon sector of the SME mework in which to explore Lorentz violation. The fra- is given by mework consists of known physics in the form of the standard model plus general relativity action along with all possible Lorentz-violating terms that can be cons- L = LEM + LCPT −odd + LCPT −even, (6) tructed from the associated fields. The mechanisms by which coefficients for Lorentz violation could arise in the SME can be divided into two categories: explicit Lorentz- where symmetry breaking and spontaneous Lorentz-symmetry breaking. The full SME includes an infinite number of 1 L = − F F µν (7) operators of ever-increasing mass dimension. A Lagran- EM 4 µν gian that describes the full SME is given as

LSME = LSM + LGR + LLV (2) is the Maxwell lagrangian, where LSM describes the usual SM fields, LGR describes 1 the usual General Relativity (GR) fields and LLV descri- κ λ µν LCPT −odd = (kAF ) κλµν A F (8) bes all possible Lorentz-violating terms constructed from 2 SM and GR fields and background coefficients. As an example, let’s consider the following Lorentz- is the CPT-odd pure-photon term, where the coupling violating terms occurring in the fermion and photon sec- κ coefficient (kAF ) is real and has dimension of mass and tors. κλµν is the totally antisymmetric Levi-Civita tensor. The CPT-even Lorentz-violating term is given by

A. Dirac ﬁeld of the SME 1 κλ µν LCPT −even = − (kF )κλµν F F , (9) The Lagrangian for the fermion sector of the extended 4 quantum electrodynamics of the SME is

↔ where the coupling (kF )κλµν is real, dimensionless and ¯ µ L = ψ iΓ ∂µ −M ψ, (3) has the same symmetry of a Riemann tensor

(kF )κλµν = −(kF )λκµν , (kF )κλµν = −(kF )κλνµ, (kF )κλµν = (kF )µνκλ, (10)

plus a double null trace which yields 19 independent com- this broad framework. An alternative procedure is to ponents. modify just the SME interaction part via a non-minimal coupling. The new interaction breaks the Lorentz and CPT symmetries. This coupling will modify the standard C. Non-minimal Lorentz-violating term Lagrangian to

Besides studies of the structure of the SME, other ideas 1 1 L = − F F µν + ψ¯ (iγµD − m) ψ − (∂ Aµ)2(11), were proposed to examine Lorentz violating operators in 4 µν µ 2ξ µ eBFIS, Ano VII ISSN:2318-8901 eBFIS 7 7241-2(2019) A. F. Santos where the covariant derivative includes a non-minimal coupling term, i.e.,

ν Dµ = ∂µ + ieAµ + igb F˜µν (12) with e, g and bµ being the electron charge, a coupling constant and a constant four vector, respectively. F˜µν = 1 αρ 2 µναρF is the dual electromagnetic tensor. There are other possible non-minimal coupling terms that exhibit Lorentz violation [17]. In this paper a Lorentz violating CPT-odd term is chosen to study Bhabha scattering. Figura 1: Exchange and annihilation diagrams with This non-minimal coupling leads to the new interaction. different vertices. The interaction lagrangian is given by ¯ µ ν ¯ µ α ρ LI = −eψγ ψAµ − gb ψγ ψ∂ A µναρ. (13) as µ µ The first term describes the usual QED vertex and the • → V = −ieγ (14) second term is a new vertex that implies violation of Lo- and rentz symmetries due to the four vector bν , which speci- ν µ α fies a preferred direction in the space-time. × → gVρ = −gb γ q µναρ, (15) where qα is the momentum operator. III. BHABHA SCATTERING WITH LORENTZ In order to calculate the cross section for this process VIOLATION the S-matriz element is written as

M = M0 + M1 + M2, (16) In this section the main results obtained in [18] are brieﬂy reviewed. Our interest is to calculate the cross where M0 is the matrix element in conventional QED, − + − + section for the process, e (p1)e (p2) → e (p3)e (p4), in M1 is linear in Lorentz-violating parameter and M2 is the presence of Lorentz-violating terms at zero tempera- quadratic in Lorentz-violating parameter. Using the La- ture. This process has the Feynman diagrams given in grangian (11) these matrix elements are given explicitly FIG. 1. These vertices in the diagrams are represented as

α α 2 u¯(p2)γ u(p1)¯v(p3)γαv(p4) u¯(p2)γ v(p4)¯v(p3)γαu(p1) M0 = e 2 − 2 , (17) (p1 − p2) (p1 + p3) σ ρ µ ν h(p1 − p2) u¯(p2)γ u(p1)¯v(p3)γ v(p4) M1 = 2egb µνσρ 2 (p1 − p2) σ ρ µ (p1 + p3) u¯(p2)γ v(p4)¯v(p3)γ u(p1)i − 2 , (18) (p1 + p3) σ τ ω 2 γ σ κλ h(p1 − p2) (p1 − p2) u¯(p2)γ u(p1)¯v(p3)γ v(p4) M2 = g b b g στλωγσκ 2 (p1 − p2) σ τ ω (p1 + p3) (p1 + p3) u¯(p2)γ v(p4)¯v(p3)γ u(p1)i − 2 . (19) (p1 + p3)

Then the diﬀerential cross section, taking an average over where |~p|2 = |p~0|2 = E2, ~p·p~0 = E2 cos θ and s = (2E)2 = 2 the spin of the incoming particles and summing over the ECM , and the relations spin of the outgoing particles, is deﬁned as X dσ 1 1 X u(p1)¯u(p1) = p/ + m, = · |M|2, (20) 1 dΩ 64π2E2 4 X CM spins v(p )¯v(p ) = p − m. (22) 1 1 /1 where ECM is the center of mass energy. Let’s consider the center of mass frame, In addition

α α p1 = (E, ~p), p2 = (E, −~p), v¯(p2)γαu(p1)¯u(p1)γ v(p2) = tr [γαu(p1)¯u(p1)γ v(p2)¯v(p2)] , 0 0 p3 = (E, p~ ) and p4 = (E, −p~ ), (21) (23) eBFIS, Ano VII ISSN:2318-8901 eBFIS 7 7241-3(2019) A. F. Santos has been used. Henceforth the electron mass is ignored mass. ν since all the momenta are large compared to the electron Considering b = (b0, 0), a time-like four vector, the diﬀerential cross section, eq. (20), is

4 2 2 2 2 2 θ dσ e (cos 2θ + 7) b0 e g sin 2 = 2 2 2 + 2 2 (−65 cos θ + 6 cos 2θ + cos 3θ + 122), (24) dΩ 256π ECM (cos θ − 1) 256π (cos θ − 1)

with the first term being the usual QED differential cross B. Bosons section at lowest order and the second term contain the contributions due to Lorentz violation. Using experimen- † By considering ap and ap being the creation and an- tal data for this scattering, that are readily available, an nihilation operators respectively, the Bogoliubov trans- upper bound to Lorentz violating parameter has been de- formation are termined. Some details about this calculation has been 0 0 † done in [18]. ap = u (β)ap(β) + v (β)ãp(β), (29) The next step is to determine the thermal corrections † 0 † 0 ap = u (β)ap(β) + v (β)ãp(β), (30) to the Bhabha scattering. The temperature effects will 0 0 † be introduced using the TFD formalism. a˜p = u (β)ãp(β) + v (β)ap(β), (31) † 0 † 0 a˜p = u (β)ãp(β) + v (β)ap(β), (32) where u0(β) = cosh θ(β) and v0(β) = sinh θ(β). IV. THERMO FIELD DYNAMICS - TFD An important note, the commutation or anti- commutation relations for creation and annihilation ope- In the TFD formalism the thermal average of any operators are similar to those at zero temperature. rator A is equal to its temperature dependent vacuum expectation value, i.e., hAi = h0(β)|A|0(β)i, where |0(β)i V. BHABHA SCATTERING AT FINITE is a thermal vacuum, β = 1 with T being the tempe- kB T TEMPERATURE rature and kB is the Boltzmann constant. Such construc- tion leads to the doubling the degrees of freedom in a Hil- bert space accompanied by the temperature dependent In this section the Bhabha scattering with Lorentz vi- Bogoliubov transformation. This doubling is defined by olation at finite temperature is studied. The Lorentz- the tilde conjugation rules, associating each operator in violating parameter is introduced via non-minimal cou- S to two operators in S , where the expanded space is pling term in the covariant derivative. In such study the T Lagrangian (11) and eq. (12) will be used. ST = S ⊗ S˜, with S being the standard Fock space and S˜ the tilde space. The thermal quantities are introduced The cross section at finite temperature is defined as through the Bogoliubov transformation which applies a dσ 1 1 X = · |M(β)|2, (33) rotation in the tilde and non-tilde variables. Let’s consi- dΩ 64π2E2 4 der the Bogoliubov transformation for fermions and bo- β CM spins sons separately. where M(β) is the S-matrix element at finite temperature, which is calculated as M(β) = hf, β|Sˆ(2)|i, βi, (34) A. Fermions with Sˆ(2) being the second order term of the Sˆ-matrix. † The Sˆ-matrix is defined as Considering cp and cp being creation and annihilation ∞ operators respectively, the Bogoliubov transformations X (−i)n Z h i Sˆ = dx dx ··· dx Hˆ (x )Hˆ (x ) ··· Hˆ (x ) , for fermions are n! 1 2 nT I 1 I 2 I n n=0 † (35) cp = u(β)cp(β) + v(β)˜cp(β), (25) † † ˆ cp = u(β)cp(β) + v(β)˜cp(β), (26) where T is the time ordering operator and HI (x) = H (x) − H˜ (x) describes the interaction. The thermal c˜ = u(β)˜c (β) − v(β)c† (β), (27) I I p p p states are † † c˜ = u(β)˜c (β) − v(β)cp(β), (28) p p |i, βi = c† (β)d† (β)|0(β)i, p1 p2 where u(β) = cos θ(β) and v(β) = sin θ(β). |f, βi = c† (β)d† (β)|0(β)i, (36) p3 p4 eBFIS, Ano VII ISSN:2318-8901 eBFIS 7 7241-4(2019) A. F. Santos with c† (β) and d† (β) being creation operators. The transition amplitude becomes pj pj

(−i)2 Z M(β) = d4x d4yhf, β|(L L − L˜ L˜ )|i, βi 2! I I I I = M0(β) + Mb(β) + Mbb(β) − M˜ 0(β) + M˜ b(β) + M˜ bb(β) , (37)

where M0(β) is the matrix element in conventional QED trix elements for the Lorentz violating interaction, res- and Mb(β) and Mbb(β) are the linear and quadratic ma- pectively. These matrix elements are given as

(0) (β) (0) There are similar equations for matrix elements that in- with ∆µν (q, β) = ∆µν (q) + ∆µν (q), where ∆µν (q) and clude tilde operators. (β) ∆µν (q) are zero and finite temperature parts respecti- By taking that the fermion field is written as vely. Explicitly Z h −ipx † ipxi ψ(x) = dp Np cpu(p)e + dpv(p)e , (41) ηµν 1 0 ∆(0)(q) = , (43) µν q2 0 −1 where Np is the normalization constant, and the photon 2 βq0/2 propagator at finite temperature, that is defined as (β) 2πiδ(q ) 1 e ∆µν (q) = − βq /2 ηµν . eβq0 − 1 e 0 1 Z 4 d q −iq(x−y) h0(β)|TAµ(x)Aν (y)|0(β)i = i e ∆µν (q, β), (2π)4 Then the Lorentz invariant transition amplitude, in the (42) center of mass frame, becomes

2h µ 0 ν M0(β) = −ie u¯(p1)γ u(p3)¯v(p4)γµv(p2)∆ (p3 − p1) − u¯(p2)γ v(p1) i βE × v¯(p )γ u(p )∆0(p + p ) tanh2 CM . (44) 3 ν 4 1 2 2

In a similar way the linear term in the Lorentz violating parameter becomes

ν h σ ρ µ 0 Mb(β) = 2egb µνσρ (p3 − p1) u¯(p1)γ u(p3)¯v(p4)γ v(p2)∆ (p3 − p1) i βE + (p + p )σu¯(p )γρv(p )¯v(p )γµu(p )∆0(p + p ) tanh2 CM , (45) 1 2 2 1 3 4 1 2 2

and the quadratic term in the Lorentz violating coeﬃci- ents is eBFIS, Ano VII ISSN:2318-8901 eBFIS 7 7241-5(2019) A. F. Santos

2 ν ρ σγ h α δ µ ω Mbb(β) = ig b b η µνασωρδγ q1 q1u¯(p1)γ u(p3)¯v(p4)γ v(p2) i βE × ∆0(p − p ) − qαqδu¯(p )γµv(p )¯v(p )γωu(p )∆0(p + p ) tanh2 CM , (46) 3 1 2 2 2 1 3 4 1 2 2

where q1 = p3 − p1 and q2 = p1 + p2. The results for In order to calculate the differential cross section, let’s ν the transition amplitudes obtained in [18] are recovered use the eqs. (22) and (23) and to consider b = (b0, 0), a 2 in the limit T → 0, which implies tanh (βECM /2) → 1 time-like four vector. Then the differential cross section, and (eβECM − 1)−1 → 0. eq. (33), at finite temperature is

" dσ e4(cos 2θ + 7)2 = 2 2 2 dΩ β 256π ECM (cos θ − 1) # b2 e2g2 sin2 θ βE + 0 2 (−65 cos θ + 6 cos 2θ + cos 3θ + 122) tanh4 CM 256π2(cos θ − 1)2 2 1 h 2 i 4 βECM + 2 2 Π1(β) + b0 Π2(β) × tanh , (47) 256π ECM 2

where

2 4 4 64π e E h 2 2 Π1(β) = (11 + 4 cos θ + cos 2θ)δ −2E (1 − cos θ) (eβECM − 1)2 i + (6 + 2 cos 2θ)δ2 4E2 + 16 cos4 (θ/2) δ −2E2(1 − cos θ) δ 4E2 ,

2 2 2 6 2 256π e g E sin (θ/2)h 2 2 2 Π2(β) = (cos 2θ + 32 sin (θ/2) + 15)δ −2E (1 − cos θ) (eβECM − 1)2 + 8 cos4 (θ/2) δ −2E2(1 − cos θ) δ 4E2 . (48)

Note that, the propagator at finite temperature introdu- that modifies the interaction part using a non-minimal ces product of delta functions with identical arguments coupling. Here a CPT-odd non-minimal coupling term is [19–22]. This problem may be avoided by working with considered. Then the effect of Lorentz violation at zero the regularized form of delta-functions and their deriva- and finite temperature on Bhabha scattering are inves- tives [23]. tigated. The TFD formalism is used to introduce the Therefore corrections for Bhabha scattering due to Lo- temperature effects. It is important to observe that the rentz violation are altered at finite temperature. Further- results give us a reasonable estimate of the Lorentz vio- more the temperature effect is very large at very high lating operators in the SME at high temperatures on the temperature, then these corrections are relevant. For Bhabha Scattering. more details about these calculations, see [24].

ACKNOWLEDGMENTS VI. CONCLUSION This work by A. F. S. is supported by CNPq projects The Standard Model Extension is an eﬀective theory 308611/2017-9 and 430194/2018-8. It is a pleasure to that allows to study violations of Lorentz and CPT sym- thank the Prof. Faqir C. Khanna for his collaboration in metries. Besides this theory, there is an alternative way the original results at ﬁnite temperature.

[1] V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 359, 545 (1991); Phys. Rev. D 51, 3923 (1995). (1989); V. A. Kostelecky and R. Potting, Nucl. Phys. B eBFIS, Ano VII ISSN:2318-8901 eBFIS 7 7241-6(2019) A. F. Santos

[2] D. Colladay, V. A. Kostelecky, Phys. Rev. D 55, 6760 Amsterdan, (1982); F. C. Khanna, A. P. C. Malbouis- (1997). son, J. M. C. Malboiusson and A. E. Santana, Themal [3] D. Colladay, V. A. Kostelecky, Phys. Rev. D 58, 116002 quantum field theory: Algebraic aspects and applicati- (1998). ons, World Scientific, Singapore, (2009). [4] H. J. Bhabha, Proc. Roy. Soc. (London) 154 A, 195 [14] H. Umezawa, Advanced Field Theory: Micro, Macro and (1936). Thermal Physics, AIP, New York, (1993). [5] K. Abe et al. (VENUS Collaboration), J. Phys. Soc. Jpn. [15] A. E. Santana and F. C. Khanna, Phys. Lett. A 203, 68 56, 3767 (1987). (1995). [6] I. Adachi et al. (TOPAZ Collaboration), Phys. Lett. B [16] A. E. Santana, F. C. Khanna, H. Chu, and C. Chang, 200, 391 (1988); AMY Collaboration, S. K. Kim et al., Ann. Phys. 249, 481 (1996). ibid. 223, 476 (1989). [17] Y. Ding and V. A. Kostelecky, Phys. Rev. D 94, 056008 [7] M. Derrick et al, Phys. Rev. D 34, 3286 (1986). (2016). [8] T. Arima et al, Phys. Rev. D 55, 19 (1997). [18] B. Charneski, M. Gomes, R. V. Maluf and A. J. da Silva, [9] V. A. Kostelecky and M. Mewes, Phys. Rev. D 80, 015020 Phys. Rev. D 86, 045003 (2012). (2009). [19] A. Das and A. Karev, Phys. Rev. D 36, 623 (1987). [10] T. Matsubara, Prog. Theor. Phys. 14, 351 (1955). [20] L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974). [11] J. Schwinger, J. Math. Phys. 2, 407 (1961); J. Schwinger, [21] A. Das and J. Frenkel, Phys. Lett. B 696, 556 (2011). Lecture Notes of Brandeis University Summer Institute [22] A. Das and J. Frenkel, Phys. Lett. B 704, 85 (2011). (1960). [23] Ch.G. van Weert, An Introduction to Real- and [12] Y. Takahashi and H. Umezawa, Coll. Phenomena 2, 55 Imaginary-time Thermal Field Theory, Lecture notes on (1975); Int. Jour. Mod. Phys. B 10, 1755 (1996). Statistical Field Theory (2001). [13] Y. Takahashi, H. Umezawa and H. Matsumoto, Thermo- [24] A. F. Santos and Faqir C. Khanna, Phys. Rev. D 95, field Dynamics and Condensed States, North-Holland, 125012 (2017).

eBFIS, Ano VII ISSN:2318-8901 eBFIS 7 7241-7(2019)