CP Asymmetries and Higher-Order Unitarity Relations
CCP Asymmetries and Higher-Order Unitarity Relations
Tom´aˇs Blaˇzek ∗ and Peter Mat´ak † Department of Theoretical Physics, Comenius University,Mlynsk´a dolina, 84248 Bratislava, Slovak Republic (Dated: February 12, 2021)CP asymmetries in particle reactions are traditionally calculated in terms of complex couplings,Feynman integrals, and Cutkosky rules. We use an expansion of the S -matrix unitarity conditioninstead, obtaining a general expression for the asymmetries without reference to the imaginary partof the loops. Asymmetry relations implied by CPT and unitarity are manifested in a diagrammaticway at any order of perturbation theory. We demonstrate the power of this general framework withinthe right-handed neutrino and top-quark scattering asymmetries in seesaw type-I leptogenesis. Introduction.
In physics, cancelations among terms incalculations may occasionally lead to small numbers orexact zero results that do not show up until the calcu-lations’ last steps are done. If this happens it may besuggestive that the language we use – the way we cutreality into pieces [1] – is not fitting the description ofthe phenomenon. We argue that this may be the caseof the S -matrix unitarity constraints relevant for contri-butions to particle-antiparticle asymmetries in the earlyuniverse. Such processes may include asymmetric decaysof heavy particles [2–4], scatterings [5, 6], annihilations[7] or multiparticle (3 ↔
2) reactions [8]. They appearin different phenomenological contexts, such as baryoge-nesis [2, 3], leptogenesis [4, 5] or asymmetric dark mat-ter models [9]. Interestingly enough, often the processescontaining different particles, or even the processes of adifferent types, conspire together to cancel each other’scontribution avoiding asymmetry generation in thermalequilibrium due to unitarity and CPT invariance [10–13].In type-I seesaw leptogenesis [14–17], the asymmetrymay come from heavy right-handed neutrino decays intolepton and Higgs boson, N ↔ lH (¯ l ¯ H ). However, theeffect of the asymmetric lH ↔ ¯ l ¯ H scattering has to betaken into account as well canceling the asymmetry inequilibrium. This is an example of what is known asa real intermediate state subtraction introduced in orderto avoid double-counting in the Boltzmann collision term[2]. Here our primary goal is to make such cancelationsas evident as possible at any order of the perturbationtheory. Moreover, we introduce a diagrammatic repre-sentation of asymmetries and reaction rates and formu-late a general algorithm identifying the complete set ofcontributions.We note that, to a certain extent, a similar approachhas been used in [14]. There, the contribution to theasymmetry of a reaction may be represented by a cyclicdiagram cut into three pieces by in , out and Cutkosky cuts. This, however, does not work that simple when anyof the three parts contains on-shell intermediate particlesor loop integrals with a non-zero imaginary part, or, interms of the transition matrix introduced bellow when T ∗ nm (cid:54) = T mn for any of the three parts. In contrast, ourapproach is fully general and can be applied both to real intermediate state subtractions and higher-order termswith complex loops. CP violation and S -matrix expansion. The S -matrixunitarity condition SS † = S † S = 1 written in terms ofthe transition matrix T fi = i( δ fi − S fi ) = (2 π ) δ (4) ( p f − p i ) M fi for the initial state | i (cid:105) and the final state | f (cid:105) leadsto the relation T † fi = T fi − i (cid:88) n T † fn T ni . (1)Then, within a CPT symmetric quantum theory, the CPasymmetry ∆ | T fi | = | T fi | − | T if | can be written as [2]∆ | T fi | = 2Im (cid:88) n T in T † nf T fi − (cid:12)(cid:12)(cid:12) (cid:88) n T † fn T ni (cid:12)(cid:12)(cid:12) . (2)Now, let us replace each T † in Eq. (2) using Eq. (1).Repeating the procedure iteratively we obtain an expan-sion∆ | T fi | = (cid:88) n (i T in i T nf i T fi − i T if i T fn i T ni ) (3) − (cid:88) n,m (i T in i T nm i T mf i T fi − i T if i T fm i T mn i T ni )+ . . . . Alternatively, the unitarity condition (1) may be useddirectly to derive∆ | T fi | = i T if ( T S † ) fi − ( S † T ) if i T fi . (4)This compact expression is equivalent to Eq. (3) by theexpansion of S † = S − as a geometric series. We canobserve that, unlike in Eq. (2), the cancelation of asym-metries in Eq. (3) after the summation over the finalstates is evident from the opposite signs of the mirroredterms in brackets. Moreover, there are no conjugatedamplitudes. A single term in Eq. (3) may be viewed asa transition from | i (cid:105) to | i (cid:105) with two or more on-shell in-termediate states including | f (cid:105) . To generate all the con-tributions of this type we introduce their diagrammaticrepresentation. First, we write forward scattering ampli-tudes in terms of Feynman diagrams to a certain order in a r X i v : . [ h e p - ph ] F e b coupling constants. Then, we make all the cuts that arekinematically allowed (see Fig. 1 bellow). Any subset oftwo or more cuts, with at least one vertex in each piece,will represent a contribution to the asymmetry. However,it is important to note that (unlike the Cutkosky cuts)here cutting refers to simply putting certain particles ona mass shell, without changing the sign of the i (cid:15) in anypropagator on any side of the cut. Asymmetries in Boltzmann collision integral.
In ma-ny scenarios of matter generation in the expanding uni-verse, the Boltzmann equation gives a reasonably accu-rate approximation to the final relic densities and asym-metries. Here we consider the particles interacting as ifthey appear in a vacuum, using the Maxwell-Boltzmannstatistics. The generalization, including the correct sta-tistical factors for the final states, can be found followingthe discussion in [11, 18]. However, before we start withthe kinetic description of particle interactions, we needto find an efficient way to deal with the products of tran-sition matrices as they appear in Eq. (3). There, thesummation includes all the combinations of distinct par-ticle species allowed by symmetries and kinematics. Themomentum integration and the summations over the dis-crete degrees of freedom are included as well. From nowon, we change this notation. The sum will only run over different particles in the intermediate states. The mo-mentum integrals and spin summations will be indicatedby the cut product defined as (cid:88) ∀ s n (cid:90) (cid:89) ∀ p n [ d p n ]i T in i T nf def . = i T in | i T nf (5)using [ d p n ] = d p n / ((2 π ) E n ) for the Lorentz invariantmeasure in the momentum space.Now, let us consider the equilibrium contribution of the i → f reaction to the evolution of the number densitiesof the included particles. Within the classical Boltzmannapproach, it is given in terms of the thermally averagedrate of the process γ eq fi = − V (cid:88) ∀ s i (cid:90) (cid:18) (cid:89) ∀ p i [ d p i ] f eq i ( p i ) (cid:19) i T † if | i T fi , (6)where V denotes the four-dimensional volume [22].Looking at Eq. (6), we can observe its trace-like struc-ture. Indeed, in equilibrium, cyclicity is guaranteed bythe detailed balance condition [19]. Thus, we can definethe trace of the product of transition matricesFr { i T in | i T nm | . . . | i T fi } def . = 1 V (cid:88) ∀ s i (cid:90) (cid:18) (cid:89) ∀ i [ d p i ] f eq i ( p i ) (cid:19) i T in | i T nm | . . . | i T fi , (7)where we have changed Tr (trace) to a new symbol Fremphasizing the presence of the equilibrium phase spacedensities (as we compute traces over the forest of Feyn-man diagrams). Using the expansion (3) we obtain forthe equilibrium asymmetry of the i → f reaction rate∆ γ eq fi = (cid:88) n Fr { i T in | i T nf | i T fi } − m . t . (8) − (cid:88) n,m Fr { i T in | i T nm | i T mf | i T fi } − m . t . + . . . with m.t. representing the mirrored terms , in which theintermediate states appear in reversed order.As an elementary example, let us consider the asym-metric right-handed neutrino decays mentioned earlier.We use the Lagrangian density L ⊃ − M i ¯ N i N i − (cid:16) Y αi ¯ N i P L l α H + H . c . (cid:17) (9)where i and α are family indices labeling the right-handedneutrino and standard model leptons, respectively. The products of diagrams depicted in Fig. 1a and 1b (minusthe mirrored terms) represent the self-energy and vertexpart of the N → lH decay asymmetry, respectively. Onthe other hand, the diagram in Fig. 1c enters the asym-metry of the lH → ¯ l ¯ H scattering. It is equivalent to the s -channel part of the real intermediate state subtractedscattering rate in the narrow width approximation [16].Moreover, having the cyclicity of the trace in mind, wecan immediately see it to be equal to Fig. 1a. The same,up to the sign, applies to Fig. 1d corresponding to theinverse decay process. Therefore, at this order in theneutrino Yukawa coupling, we only have two indepen-dent contributions entering the relevant processes’ asym-metries - those containing the s - or t -channel neutrinopropagator in the scattering diagram depicted in Fig. 1aand 1b. Everything else may be expressed in terms oftheir cyclic permutations and conjugations. Neutrino-quark scattering and O ( Y Y t ) unitarity re-lations. The lowest order asymmetries are simple, evenwithin the standard Cutkosky approach. Here, as a moreadvanced example, we consider the scattering of right-handed neutrino and top-quark [8, 20]. To this purpose, l α H N j ¯ H ¯ l β N i N i (a) l α H N j ¯ H ¯ l β N i N i (b) N j ¯ H ¯ l β N i Hl α Hl α (c) N i ¯ l β ¯ H N j Hl α Hl α (d) FIG. 1: Lepton number violating contributions to ∆ | T ( N → lH ) | (Fig. 1a, 1b), ∆ | T ( lH → ¯ l ¯ H ) | (Fig. 1c) and ∆ | T (¯ l ¯ H → N ) | (Fig. 1d). For the scattering and inverse decay additional t -channel diagrams must be included as well. QH tN j Hl α N i Hl β (a) QH t Hl α N i l β HN j (b) FIG. 2: Lepton number violating vacuum diagrams leadingto asymmetries at the O ( Y Y t ) order. we add to the Lagrangian density (9) L ⊃ −Y t ¯ tP L QH + H . c ., (10)with t and Q representing the right-handed top and left-handed quark doublet, respectively. All particles, exceptfor the right-handed neutrinos, are considered massless.As we already mentioned, given the initial state N i Q weshould take all the corresponding forward scattering di-agrams and cut them into three, four, or as many piecesas possible. What particular vacuum diagrams shouldwe start with? To obtain a reliable answer, we start withall vacuum bubbles made out of four neutrino and twotop-quark Yukawa vertices. There are six of them. Onecomes with |Y αi | |Y βj | Y t , a real combination of cou-plings that does not affect the asymmetry. Three bub-bles can be made in a lepton number conserving way, proportional to Y ∗ αi Y αj Y βi Y ∗ βj Y t . The contribution ofthese vanishes after summing through the lepton flavors.Finally, there are two lepton number violating vacuumdiagrams shown in Fig. 2. There is one Q propagatorin each of them and one way of cutting it to obtain the N i Q → N i Q diagram. There are two right-handed neu-trino propagators and two ways of cutting each. Theresulting diagrams are shown in Fig. 3, where some ofthe family indices have been relabeled so that i corre-sponds to the initial right-handed neutrino while α and β label the lepton and antilepton, respectively [23]. Togenerate the T ’s of Eq. (8) we consider all possible cutsof these diagrams into three or more pieces. Each piecestands for one amplitude T . Note that T fi , the first pieceof each diagram, defines the process.We are now ready to consider the relevant ways of cut-ting the diagram Fig. 3a. Four different cuts can bemade, but only a certain combination of these can af-fect the asymmetry. If, for example, the ¯ l β ¯ H loop on theright-handed neutrino leg remains uncut, there is onlyone way to split this diagram into three pieces (see Eq.(8)). It requires two cuts of the l α line, each cutting oneof the Higgs box propagators. The corresponding mir-rored terms add to the asymmetry of the same process, N i Q → lHQ . However, they are completely canceled outby cuts of the diagram in Fig. 3e. If, on the other hand,the ¯ l β ¯ H loop is cut, we get contributions to the asym-metries of N i Q → lt , N i Q → lHQ and, with a minussign, to the mirrored terms of N i Q → ¯ l ¯ HQ . Now, cut-ting the diagram in Fig. 3e will add to the asymmetryof N i Q → ¯ l ¯ HQ and the mirrored terms of N i Q → lt , N i Q → lHQ . Using the expansion of the reaction rateasymmetry in Eq. (8), we can write l α H N j ¯ l β ¯ Ht HQN i QN i (a) l α H N j ¯ l β HH tQN i QN i (b) l α H N j ¯ l β ¯ Ht HQN i QN i (c) l α H N j ¯ l β HH tQN i QN i (d)¯ l β N j ¯ H l α H t HQN i QN i (e) ¯ l β N j H l α HH tQN i QN i (f) ¯ l β N j H l α ¯ H t HQN i QN i (g) ¯ l β N j H l α HH tQN i QN i (h) FIG. 3: Forward scattering diagrams for the N i Q initial state obtained from Fig. 2. Cutting them will generate asymmetryrelations for reactions with two-particle color triplet initial states at the O ( Y Y t ) order. ∆ γ ( a ) N i Q → lt = Fr H l α t N j H Q ¯ H ¯ l β N i Q N i Q − m . t ., (11a)∆ γ ( a ) N i Q → lHQ = Fr H l α t QH QN j ¯ H ¯ l β N i Q N i Q + Fr l α H Q N j HQt ¯ H ¯ l β N i Q N i Q (11b) − Fr Hl α N i Q l α Q t QH QN j H ¯ H ¯ l β N i Q − m . t ., ∆ γ ( a ) N i Q → ¯ l ¯ HQ = − ∆ γ ( a ) N i Q → lt − ∆ γ ( a ) N i Q → lHQ . (11c)Splitting the uncut Higgs propagators in Eq. (11b) intothe principal value and Dirac delta function cancels thelast singular term. This feature, however, is not generic,and there are examples of diagrams where (non-singular)higher-order terms of the expansion (8) persist. The re-maining terms in Eq. (11b) no longer contain i (cid:15) in prop-agators, and the resulting asymmetry is proportional tothe couplings’ imaginary part. Repeating the procedurepairwise for all the diagrams in the upper and lower rowsof Fig. 3 (and sorting them out according to reactions,namely by the first piece) leads straightforwardly to the O ( Y Y t ) unitarity relation0 = ∆ γ eq N i Q → lt + ∆ γ eq N i Q → lHQ (12)+ ∆ γ eq N i Q → ¯ l ¯ HQ + ∆ γ eq N i Q → ¯ lQQ ¯ t . For the sake of simplicity, the processes occurring abovethe M j threshold have been neglected in Eq. (12). Anal-ogous relations may be written for the N i ¯ t initial stateusing the apparent Q ↔ ¯ t symmetry of Fig. 2 or, usingEq. (3), for unaveraged asymmetries ∆ | T fi | instead ofthermally averaged ∆ γ eq fi .Before we conclude, let us comment briefly on the in-terpretation of the 2 → → Summary.
Using the expansion of the S -matrix uni-tarity condition we have obtained a diagrammatic repre-sentation of the CP asymmetries in ∆ | T fi | and ther-mally averaged reaction rates ∆ γ eq fi . We generate acomplete set of contributions to the reaction’s asym-metries starting with cuts of the vacuum diagrams atparticular order in coupling constant. Unlike the stan-dard Cutkosky approach that uses a single cut to de-termine the loop’s imaginary part we introduce multiplecuts of the corresponding forward scattering diagrams.With this procedure the cancelations among the result-ing asymmetries are easy to track. Our general frame-work was demonstrated in an analysis of the higher-ordercorrections to the right-handed neutrino scatterings inseesaw type-I leptogenesis. Acknowledgements.
We would like to thank our col-league, Vladim´ır Balek, for useful comments and discus-sion. The authors were supported by the Slovak Ministryof Education five-year contract 0211/2016. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] L. Kvasz,