PHYSICAL REVIEW D 103, 056015 (2021)
Field dependence of the Yukawa coupling in the three flavor quark-meson model
† G. Fejős * and A. Patkós Institute of Physics, Eötvös University, 1117 Budapest, Hungary
(Received 16 November 2020; accepted 23 February 2021; published 22 March 2021)
We investigate the renormalization group flow of the field-dependent Yukawa coupling in the framework of the three flavor quark-meson model. In a conventional perturbative calculation, given that the field rescaling is trivial, the Yukawa coupling does not get renormalized at the one-loop level if it is coupled to an equal number of scalar and pseudoscalar fields. Its field-dependent version, however, does flow with respect to the scale. Using the functional renormalization group technique, we show that it is highly nontrivial how to extract the actual flow of the Yukawa coupling as there are several new chirally invariant operators that get generated by quantum fluctuations in the effective action, which need to be distinguished from that of the Yukawa interaction.
DOI: 10.1103/PhysRevD.103.056015
I. INTRODUCTION the fermions and the field strength renormalization is trivial [6]. In phenomenological investigations of the two and One of the merits of the functional renormalization group three flavor quark-meson models, several papers have dealt (FRG) technique is that it allows for calculating the flows of with the nonperturbative renormalization of the Yukawa n-point functions nonperturbatively. Via the FRG one has coupling. The essence of the corresponding calculations is the freedom to evaluate the scale dependence of them in that one determines the RG flow of the fermion-fermion- nonzero background field configurations, which is thought meson proper vertex, defined as δ3Γ=δ ψδ¯ ψδϕ in a given to be essential once spontaneous symmetry breaking occurs L R background (Γ being the effective action), and then [1–3]. For the sake of an example, in scalar ðϕÞ theories, associates it with the flow of the Yukawa coupling itself. quantum corrections to the wave function renormalization This obviously works for one flavor models [7,8], and even (Z) typically vanish at the one-loop level, but using the for the two flavor case [9,10], in particular for models FRG one gets a nonzero contribution once one generalizes restricted to the σ − π subsector, i.e., with Oð4Þ symmetry the corresponding kinetic term in the effective action as μ in the meson part of the theory. The reason why the ∼Z ϕ ∂μϕ∂ ϕ Z ð Þ and evaluates at a symmetry breaking approach has legitimacy in the latter case is that it turns out stationary point of the effective action. This procedure is that the only way fluctuations generate new Yukawa-like essential, e.g., in two-dimensional systems that undergo operators in the quantum effective action is that the original topological phase transitions, as the wave function renorm- Yukawa term is multiplied by powers of the quadratic Oð4Þ alization is known to be diverging in the low temperature invariant of the meson fields. That is to say, one is allowed phase, which cannot be described in terms of perturbation to choose, e.g., a vacuum expectation value for which all theory [4]. Similarly, in four dimensions, it has recently been pseudoscalar (π) fields vanish, but the scalar (s) one is shown that in the three flavor linear sigma model the nonzero, and associate ∼s2 with the aforementioned quad- ‘ ’ coefficient of t Hooft s determinant term also receives ratic invariant leading to the appropriate determination of substantial contributions when evaluated at nonzero field [5]. the field dependence of the Yukawa coupling. A similar treatment should be in order for Yukawa The situation is not that simple in case of three flavors, interactions, whose renormalization group flow vanishes i.e., chiral symmetry of ULð3Þ × URð3Þ. Since the effective at the one-loop level, if complex scalar fields are coupled to action automatically respects linearly realized symmetries of the Lagrangian, field dependence of any coupling must *[email protected] mean all possible functional dependence on chirally invari- † [email protected] ant operators. Because there are more invariants compared to the two flavor case, if one chooses a specific background Published by the American Physical Society under the terms of for the evaluation of the RG flow of the fermion-fermion- the Creative Commons Attribution 4.0 International license. “ ” Further distribution of this work must maintain attribution to meson vertex, one may arrive at an effective Yukawa the author(s) and the published article’s title, journal citation, interaction, but it will then mix in that specific background and DOI. Funded by SCOAP3. the contribution of different chirally invariant operators.
2470-0010=2021=103(5)=056015(13) 056015-1 Published by the American Physical Society G. FEJŐS and A. PATKÓS PHYS. REV. D 103, 056015 (2021)
The reason is that for three flavors other operators than that The paper is organized as follows. In Sec. II, we present the of the Yukawa term (multiplied by powers of the quadratic model emphasizing its symmetry properties. We explicitly invariant) can get generated, which are presumably not show what kind of new operators can be generated and set an even considered neither in the Lagrangian nor in the ansatz ansatz for the scale-dependent effective action accordingly. of the effective action. The newly emerging operators in Section III contains the explicit calculation of the flows principle cannot be distinguished from the Yukawa inter- generated by the field-dependent Yukawa coupling, and action if the background is too simple, and as it will be Sec. IVis devoted to an extended scenariowhere at the lowest shown, a naive calculation squeezes inappropriate terms order the newly established operators are also taken into into the Yukawa flow, which should be separated via the account. In Sec. V, we present numerical evidence for the choice of a suitable background and projected out. relevance of the proposed procedure, while the reader finds Investigations of the three flavor case including the the summary in Sec. VI. effects of the UAð1Þ anomaly were started by Mitter and Schaefer. In a first paper, no RG evolution of the Yukawa II. MODEL AND SYMMETRY PROPERTIES coupling was taken into account [11]. This approximation We are working with the U ð1Þ anomaly free three flavor was carried over in several directions, including inves- A quark-meson model, which is defined through the follow- tigations regarding magnetic [12] and topological [13] ing Euclidean Lagrangian: susceptibilities, mass dependence of the chiral transition [14], or quark stars [15]. † 2 † L ¼ Trð∂iM ∂iMÞþm TrðM MÞ A strategy for extracting the RG flow of the field- λ¯ † 2 λ¯ † † dependent Yukawa coupling was presented by Pawlowski þ 1ðTrðM MÞÞ þ 2TrðM MM MÞ and Rennecke, explicitly realized for the two flavor case þ q¯ð=∂ þ gM5Þq; ð1Þ [10]. In this paper, interaction vertices arising from the π expansion of the field-dependent Yukawa coupling were where M stands for the meson fields, M ¼ðsa þ i aÞTa λ 2 3 λ also analyzed and it was found that higher order mesonic (Ta ¼ a= are generators of Uð Þ with a being the Gell- interactions to two quarks are gradually less important. Mann matrices, a ¼ 0…8), qT ¼ðudsÞ are the quarks, and 2 These results were further used in recent ambitious com- M5 ¼ðs þ iπ γ5ÞT .Asusual,m is the mass parameter ¯ ¯ a a a putations of the meson spectra from effective models and λ1, λ2 refer to independent quartic couplings. The fermion directly matched to QCD [16,17]. The method of [10] part contains =∂≡∂iγi, γ5 ¼ iγ0γ1γ2γ3 with fγig being the was then applied for extracting the flow of the Yukawa Dirac matrices and the Yukawa coupling is denoted by g. coupling in the 2 þ 1 flavor (realistic) quark-meson model Concerning the meson fields, ULð3Þ × URð3Þ ≃ [18]. The prescription used in the latter calculation relies on UVð3Þ × UAð3Þ chiral symmetry manifests itself as a symmetry breaking background, from which a field- † † † independent Yukawa coupling is extracted. A similar M → VMV ;M→ A MA ð2Þ approach has already been introduced in the seminal paper of Jungnickel and Wetterich [19], where a diagonal for vector and axialvector transformations, respectively, θa θa symmetry breaking background was employed in calculat- V ¼ expði VTaÞ, A ¼ expði ATaÞ. This also implies that ing the RG equation of the Yukawa coupling (for general † † † number of flavors). As a generalization, here we offer a M5 → VM5V ;M5 → A5M5A5; ð3Þ renormalization procedure of the field-dependent Yukawa θa γ coupling realizable in any symmetry breaking background where A5 ¼ expði ATa 5Þ. As for the quarks, the trans- through a chirally invariant set of operators. formation properties are More specifically, the aim of this paper is to show how to separate the field-dependent Yukawa interaction from those q → Vq; q → A5q: ð4Þ new fermion-fermion-multimeson couplings that unavoid- γ γ 0 ably arise as three flavor chiral symmetry allows for a much Note that, since f i; 5g¼ , larger set of operators in the infrared than considering only † q¯ → qV¯ ; q¯ → qA¯ 5: ð5Þ a field-dependent Yukawa coupling. For this, the right-hand side (rhs) of the evolution equation of the effective action These transformation properties guarantee that all will be evaluated in such a background that allows for terms in (1) are invariant under vector and axialvector expressing it in terms of explicitly invariant operators, as it transformations. was demonstrated for purely mesonic theories some time The quantum effective action, Γ, built upon the theory ago [20–23]. We also note that apart from the above defined in (1) also has to respect chiral symmetry. That is to applications, recent studies on a lower Higgs mass bound say, only chirally invariant combinations of the fields can [24] and on quantum criticality [25] also deal with field- emerge in Γ. For three flavors, there exist three independent dependent Yukawa interactions. invariants made out of the M fields,
056015-2 FIELD DEPENDENCE OF THE YUKAWA COUPLING IN THE … PHYS. REV. D 103, 056015 (2021)
† ρ ≔ TrðM MÞ; the renormalization group flow of gðρÞ, then we need a τ ≔ TrðM†M − TrðM†MÞ=3Þ2; strategy to distinguish each operator at a given order, in order † † 3 to be able to resum only the Yukawa interactions, and not the ρ3 ≔ TrðM M − TrðM MÞ=3Þ ; ð6Þ aforementioned new quark-quark-multimeson operators. Obviously, this procedure cannot be done through a one where the last one is absent in (1) due to perturbative component background field, e.g., the vectorlike condensate UV renormalizability, but nothing prevents its generation in M ∼ s0 · 1, as it mixes up the corresponding operators, and the infrared. In principle, the chiral effective potential, V, Γ one loses the chance to recombine the field dependence into Rdefined via homogeneous field configurations, jhom ¼ actual invariant tensors. ρ τ ρ x V has to be of the form V ¼ Vð ; ; 3Þ. Obviously, the The flow of the effective action is described by the generalized Yukawa term Wetterich equation [27,28] Z gðρ; τ; ρ3Þ · qM¯ 5q ð7Þ 1 2 ∂ Γ Γð Þ R −1∂ R ; k k ¼ 2 Tr½ð k þ kÞ k k ð12Þ can (and does) also appear in Γ, but in this paper we restrict ourselves to Γð2Þ where k is the second functional derivative matrix of Γ and R is the regulator function. One typically gðρ; τ; ρ3Þ ≈ gðρÞ: ð8Þ k k chooses it to be diagonal in momentum space, and set Rk R 2π 4 δ This is motivated from a symmetry breaking point of view, as kðp; qÞ¼ð Þ RkðqÞ ðq þ pÞ. Evaluating (12) in i.e., without explicit symmetry breaking terms in (1), chiral Fourier space leads to symmetry breaks as ULð3Þ × URð3Þ → UVð3Þ [26], and for Z 1 2 any background field respecting vector symmetries, ∂ Γ Γð Þ R −1 p; −p ∂ R p : k k ¼ 2 Tr½ð k þ kÞ ð Þ k kð Þ ð13Þ τ ≡ 0 ≡ ρ3. We can think of gðρÞ as p X∞ In this paper, the equation will be evaluated only in ρ ρn ρ … gð Þ¼ gn ¼ðg0 þ g1 · þ Þ; ð9Þ spacetime-independent background fields; therefore, one 0 n¼ Γð2Þ assumes that k is also diagonal in momentum space, which shows that it actually resums operators of the Γð2Þ 2π 4Γð2Þ δ † n k ðp; qÞ¼ð Þ k ðqÞ ðq þ pÞ. In such backgrounds, form ∼½TrðM MÞ qM¯ 5q. it is sufficientR to work with the effective potential, Vk, via Note that, however, as announced in the Introduction, Γ j ¼ V , and (13) leads to there are several other invariant combinations at a given k hom x k Z order that are allowed by chiral symmetry and contribute to 1 2 ∂ V Γð Þ p −1∂ R p ; quark-meson interactions. Take the lowest nontrivial order, k k ¼ 2 Trfð k ð RÞÞ k kð Þg ð14Þ i.e., dimension 6. We have two invariant combinations, p
† where we have assumed that the regulator matrix Rk is ∼ρ · qM¯ 5q; ∼qM¯ 5M5M5q: ð10Þ meant to replace everywhere p with pR, where pR is the regulated momentum through the R function, i.e., Obviously the first one is to be incorporated into the k p2 ¼ p2 þ R ðpÞ, p ¼ p pˆ . field-dependent Yukawa coupling, but not the second one. R k Ri R i The ansatz we choose for Γ is called the local potential Note that one may think of the fifth Dirac matrix as the k approximation, which consists of the usual kinetic terms difference between left- and right-handed projectors, and a local potential (in this case, specifically, a mesonic γ5 ¼ PR − PL, and therefore, e.g., the first expression ¯ ¯ † potential plus the Yukawa term), is equivalent to qLMqR þ qRM qL and the second to Z ¯ † ¯ † † qLMM MqR þ qRM MM qL. Thus, one can trade the † Γk ¼ ½Trð∂iM ∂iMÞþq¯=∂q M5 dependence to M dependence in the invariants via x working with left- and right-handed quarks separately. þ V ðρ; τ; ρ3Þþg ðρÞqM¯ 5q : ð15Þ Going to next-to-leading order, we see the emergence of ch;k k ρ ¯ the following new terms [note that for general configura- Using the earlier notation, Vk ¼ Vch;k þ gkð ÞqM5q. Note † † tions TrðM5M5Þ¼TrðM MÞ]: that the chiral potential will not be considered in its full generality, but rather as V ¼ U ðρÞþC ðρÞτ. Flows of 2 † † † ch;k k k ∼ρ qM¯ 5q; ∼ρ qM¯ 5M M5q; ∼qM¯ 5M M5M M5q; · · 5 5 5 Uk and Ck can be found in [23]. Note that this construction could be easily extended with ‘t Hooft’s determinant term ð11Þ † describing the UAð1Þ anomaly, ∼ðdet M þ det M Þ, and the and we might even have ∼τ · qM¯ 5q, though we intend to drop corresponding coefficient function. Investigations in this such contributions; see approximation (8). If we are to extract direction are beyond the scope of this paper.
056015-3 G. FEJŐS and A. PATKÓS PHYS. REV. D 103, 056015 (2021)
III. FLOW OF THE FIELD-DEPENDENT where the mass matrices are defined through the following YUKAWA COUPLING relations: In this section, we evaluate (13) in homogeneous back- ground fields to obtain the flow of gkðρÞ, defined in (15). 2 Calculating Γð Þ using the ansatz (15), one gets the following 2 ∂2 ∂ ∂ 2 ∂2 ∂π ∂π k ðms;kÞij ¼ Vk= si sj; ðmπ;kÞij ¼ Vk= i j; hypermatrix in the fsa; πa; q¯ T;qg multicomponent space: 2 ∂2 ∂ ∂π 0 1 ðmsπ;kÞij ¼ Vk= si j: ð17Þ p2 þ m2 m2 −g ðTq⃗ ÞT g q¯ T⃗ B s;k sπ;k k k C B 2 2 2 ⃗ T ⃗ C B mπ p þ mπ −igkðγ5TqÞ igkq¯ T γ5 C B s;k ;k C; B ⃗ ⃗ C Γð2Þ @ gkTq igkγ5Tq 0 ip= þ gkM5 A For practical purposes, it is worth to separate k into three ð2Þ ð2Þ ð2Þ ð2Þ ⃗T ⃗T parts [29,30], Γ ¼ Γ þ Γ þ Γ , where the respec- −gkðq¯TÞ −igkðq¯TÞ γ5 −ip= − gkM5 0 k k;B k;F k;mix tive terms are defined as ð16Þ
0 1 0 1 2 2 2 00 00 0 0 p þ ms;k msπ;k B 2 2 2 C B C 2 B m p m 00C 2 B 00 0 0 C Γð Þ B πs;k þ π;k C Γð Þ B C k;B ¼ @ A; k;F ¼ @ A; 0000 00 0 ip= þ gkM5 0000 00−ip= − g M5 0 0 k 1 ⃗ T ⃗ 00−gkðTqÞ gkq¯ T 2 ! B C 0 Γð Þ B ⃗ T ⃗ C 2 k;FB B 00−ig ðγ5TqÞ ig q¯ T γ5 C Γð Þ ≡ ¼ B k k C: ð18Þ k;mix Γð2Þ 0 B ⃗ γ ⃗ 00C k;BF @ gkTq igk 5Tq A ⃗ T ⃗ T −gkðq¯ TÞ −igkðq¯ TÞ γ5 00
Using these notations, and by introducing the differential interested in the flow of the Yukawa coupling; thus, ˜ operator ∂k, which, by definition acts only on the regulator we need to identify in the rhs of (20) the operator function, Rk, (14) can be conveniently reformulated. ∼qM¯ 5q. Diagrammatically, the commonly known term We may use the matrix identity that contributes is seen in Fig. 1, which is generated ! from the leading term of the last contribution of the rhs Γð2Þ Γð2Þ k;B k;BF of (20), Γð2Þ Γð2Þ k;FB k;F Z ! ! 1 ð2Þ ð2Þ ð2Þ ð2Þ −1 ð2Þ − ∂˜ Γ Γ Γ 0 1 Γ Γ kTr½Gk;FðpRÞ k;FBGk;BðpRÞ k;BF : ð21Þ k;B ð k;BÞ k;BF 2 ¼ ð19Þ p Γð2Þ 1 0 Γð2Þ − Γð2Þ Γð2Þ −1Γð2Þ k;FB k;F k;FBð k;BÞ k;BF More specifically, it reads and arrive at Z 1 2 ∂ V ∂˜ Γð Þ p k k ¼ 2 kTr log k ð RÞ Zp Z 1 2 1 2 ∂˜ Γð Þ p ∂˜ Γð Þ p ¼ 2 kTr log k;Bð RÞþ2 kTr log k;Fð RÞ pZ p 1 ∂˜ 1 − Γð2Þ Γð2Þ þ kTr log½ Gk;FðpRÞ k;FBGk;BðpRÞ k;BF ; 2 p ð20Þ FIG. 1. One-loop triangle diagram that is responsible for Γð2Þ −1 Γð2Þ −1 where Gk;B ¼ð k;BÞ , Gk;F ¼ð k;FÞ and the negative flowing quark-meson interactions. The solid lines correspond sign of the pure fermionic term is understood. We are to quarks, while the dashed ones represent mesons.
056015-4 FIELD DEPENDENCE OF THE YUKAWA COUPLING IN THE … PHYS. REV. D 103, 056015 (2021)
−1 where SkðpÞ¼ðip þ gkM5Þ is the fermion propagator without doubling, while Gk;s, Gk;π, and Gk;sπ are the respective 9 × 9 submatrices of Gk;B in a purely bosonic background (see Appendix B). In principle, the evaluation of (22) goes as follows. First, one evaluates Γk;B, then inverts it to obtain Gk;s, Gk;π, and Gk;sπ. Second, one inverts the fermion propagator, and finally performs all matrix FIG. 2. One-loop tadpole diagram arising from the field- ¯ dependent Yukawa interaction, which allows the generation of multiplications to get the operator that is sandwiched by q new quark-meson vertices in the RG flow. The solid lines and q. Note that the procedure turns out to be too correspond to quarks, while the dashed ones represent mesons. complicated to be carried out in a general background, but as we explain in the next subsections, fortunately we will not need to evaluate (22) in its full generality. On top of the above, via a field-dependent Yukawa coupling there is also contribution from the first term of (20), as meson masses get modified by a fermionic back- ground (see Appendix B) leading to the possibility of generating a ∼qM¯ 5q term in the effective action; see Fig. 2. Generically, this contributes to the rhs of (20) as ð22Þ
ð23Þ
A. Flows in the symmetric phase 1 ∂2τ Gab ðpÞ¼− s π U00 þ C ; ð24cÞ k;sπ 2 0 2 a b k ∂ ∂π k As a first step, we are interested in the RG flows in the ðp þ UkÞ sa b symmetric phase, i.e., where they are obtained at zero field. † Note that, however, this still necessitates the evaluation of ip g M S ðpÞ¼− þ k 5 ; ð24dÞ the rhs of the flow equation (22) in a nonzero background, k p2 p2 but if we are interested in the flowing couplings of dimension 6 operators, it is sufficient to perform all 0 00 where the coefficient functions (i.e., Uk, Uk, Ck) are computations at the cubic order in the meson fields. evaluated at zero field. Note that in this subsection all That is to say, when all propagators are expanded around computations are performed without specifying any back- 0 M ¼ (see Appendix B for the general formulas), one is ground field and keeping its most general form. Terms that allowed to work with them at the following accuracy: contain higher derivatives of Ck are left out since their 1 1 coefficients would lead to subleading (higher than cubic Gab ðpÞ¼ δ − power) contributions in the background fields in (22) k;s p2 þ U0 ab ðp2 þ U0 Þ2 k k and (23). ∂2τ 00 Note that the scalar and pseudoscalar meson propagators × sasbUk þ Ck ; ð24aÞ do not mix in the symmetric phase and are degenerate; ∂sa∂sb therefore, there is a relative (−1) factor between their 1 1 ab contributions in (22), which exactly cancels the term linear G πðpÞ¼ δ − k; p2 þ U0 ab ðp2 þ U0 Þ2 in the mesonic fields. In the piece of the integrand that is k k ∂2τ quadratic in them, the quark momentum is odd; therefore, π π U00 C ; upon integration, it also vanishes. The leading contribution × a b k þ ∂π ∂π k ð24bÞ a b of (22) is determined by the mass terms,
056015-5 G. FEJŐS and A. PATKÓS PHYS. REV. D 103, 056015 (2021)
We close this subsection by listing the coupled flow equations for the Yukawa coupling and its deri- vative evaluated at zero field, i.e., in the symmetric phase, ∼¯ † − ρ 3 00 when qðM5M5 = ÞM5q is projected out. Dropping gk in order to close the system of equations, and by using the definitions of (9), (26), and (27), we are led to Z 1 ∂ 10 ∂˜ ð25Þ kg0;k ¼ g1;k k 2 0 ; ð28aÞ p pR þ Uk Z Exploiting various identities of the Uð3Þ algebra (see 1 16 1 3 00 3 ˜ Appendix A), one can perform the algebraic evaluation ∂ g1 ¼ g0 U þ g0 C ∂ k ;k 3 ;k k 9 ;k k k p2 ðp2 þ U0 Þ2 of the integrands to arrive at pZ R R k 16 1 00 ˜ − g1 C þ 3g1 U ∂ : 3 ;k k ;k k k 2 0 2 p ðpR þ UkÞ ð28bÞ ð26Þ B. Flows in the broken phase Now, we explore how the flowing couplings depend on the fields, more specifically, as described in (8), their ρ Equation (26) clearly shows the appearance of a new dependence will be determined. † dimension 6 operator (∼qM¯ 5M5M5q), which was absent At this point, we specify a background in which we will in the ansatz of the effective action. Before discussing this evaluate the flow equation, as in case of general fM; q;¯ qg result, let us turn to the contribution of (23). Using, again, configurations, there is no hope that the broken symmetry expressions (24), a straightforward calculation leads to phase calculations can be done explicitly. The nice thing, however, is that it is sufficient to work in a restricted background. We, again, definitely need to assume nonzero q¯, q, and then we specify the M ¼ s0T0 þ s8T8 (physical) condensate. This is one of the minimal choices, which allows † for a unique restoration of the ∼qM¯ 5M5M5q and ∼qM¯ 5q operators in the rhs of the flow equation (a one component condensate would definitely not allow to do so). We have checked explicitly with other backgrounds the uniqueness of 27 ð Þ the results, and as expected, found agreement. As outlined above, the first step is to evaluate the where the coefficient functions (i.e., U0 ;g ;g0 ;g00;C ) are, k k k k k Γð2Þ Γð2Þ once again, evaluated at zero field. The flow of the effective two-point functions k;s , k;π (note that in the current potential is the sum of (26) and (27). background there is no s − π mixing), see details in At this point, it is clear that the flow equation does not Appendix B, in particular (B10) and (B11). Then one close in the sense that the ansatz (15) does not contain the inverts these matrices to obtain Gk;s and Gk;π. They are † – operator ∼qM¯ 5M M5q; therefore, for consistency reasons, diagonal except for the 0 8 sectors. Similarly, for the 5 −1 ρ it has to be dropped also in the rhs of (20). Instead of doing fermion propagator, one has Sk ðpÞ¼ðip þ gkð ÞM5Þ, † which leads to the inverse so, one may choose to project out the ∼q¯ðM5M5 − ρ=3ÞM5q term, because in the symmetry breaking pattern † −ip þ gkM5 ULð3Þ × URð3Þ → UVð3Þ that is the actual combination, SkðpÞ¼ 2 † : ð29Þ which vanishes. This choice should lead to the appropriate p þ gkM5M5 definition of the field-dependent flowing Yukawa coupling Since in the background in question ½Sk;M5 ¼0, Eq. (22) if no explicit symmetry breaking terms are present. Note, becomes however, that, once finite quark masses and the corre- sponding explicit breaking terms are introduced, in the † minimum point of the effective potential M5M5 ≠ ρ=3, and the aforementioned projection can be regarded as some- what arbitrary. A more appropriate treatment is to include † ∼qM¯ 5M5M5q in the ansatz of the effective action in the first place (see Sec. IV). ð30Þ
056015-6 FIELD DEPENDENCE OF THE YUKAWA COUPLING IN THE … PHYS. REV. D 103, 056015 (2021)
Similarly, exploiting the choice of the simplified back- It is important to mention that one does not need to work ground, (23) becomes with general s0, s8 background values, but can safely assume that s8 ≪ s0, and thus expand both (30) and (31) in terms of s8 and work in the leading order. This simplifi- cation still uniquely allows for identifying the flow of the † ∼qM¯ 5q and ∼q¯ðM5M5 − ρ=3ÞM5q operators. The latter, as expected, pops up again [it is inherently of Oðs8Þ]; thus, a two component background is indeed necessary to obtain each flow. The sum of (30) and (31) leads to ð31Þ
Z 3 3 8 1 ˜ 0 ∂ g ðρÞ þ þ qM¯ 5q; k 2 k p2 þ U0 ðρÞ 3p2 þ 4C ðρÞρ þ 3U0 ðρÞ p2 þ U0 ðρÞþ2ρU00ðρÞ p Z R k R k k R k k 3 ρ 3 2 − 2 ρ ρ 16 ρ 2 0 ρ 3 2 12 ρ ρ 0 ρ 00 ρ ˜ g ð Þð pR g ð Þ Þð Ckð ÞðpR þ U ð ÞÞ þ ðpR þ Ckð Þ þ U ð ÞÞU ð ÞÞ þ ∂ k k k k k k ð3p2 þ g2ðρÞρÞ2ðp2 þ U0 ðρÞÞð3p2 þ 4C ðρÞρ þ 3U0 ðρÞÞðp2 þ U0 ðρÞþ2ρU00ðρÞÞ p R k R k R k k R k k 00 ρ gkð Þ þ ρqM¯ 5q; p2 þ U0 ðρÞþ2ρU00ðρÞ ZR k k 2 3 ρ 3 2 − 2 ρ ρ 2 0 ρ 2 0 ρ 2 ˜ g ð Þð pR g ð Þ ÞðA0 þ A1ðpR þ U ð ÞÞ þ A2ðpR þ U ð ÞÞ Þ þ ∂ k k k k k ð6p2 þ 2g2ðρÞρÞ3ðp2 þ U0 ðρÞÞ2ð3p2 þ 4C ðρÞρ þ 3U0 ðρÞÞ2ðp2 þ U0 ðρÞþ2ρU00ðρÞÞ p R k R k R k k R k k 6 0 ρ 3 ρ 2 0 ρ ρ gkð Þð Ckð Þþ Ckð Þ Þ † − q¯ðM5M − ρ=3ÞM5q; ð32Þ 3 2 4 ρ ρ 3 0 ρ 2 0 ρ 2ρ 00 ρ 5 ð pR þ Ckð Þ þ Ukð ÞÞðpR þ Ukð Þþ Ukð ÞÞ where
−288 3 ρ ρ2 3 2 2 ρ ρ 2 0 ρ 2ρ 00 ρ 36 2 0 ρ 3 4 ρ ρ 3 2 2 ρ ρ 9 2 00 ρ A0 ¼ Ckð Þ ð pR þ gkð Þ Þðk þ Ukð Þþ Ukð ÞÞ þ ðpR þ Ukð ÞÞ ð Ckð Þ ð pR þ gkð Þ Þþ pRUkð ÞÞ; −144 2 ρ ρ 14 2 4 2 ρ ρ 2 0 ρ 12ρ 00 3 2 2 ρ ρ A1 ¼ Ckð Þ ðð pR þ gkð Þ ÞðpR þ Ukð ÞÞ þ Ukð pR þ gkð Þ ÞÞ; −12 ρ 18 4 6 2 3 0 ρ 12ρ 00 ρ − 8 0 ρ ρ2 4 2 ρ ρ2 9 00 ρ − 4 ρ ρ A2 ¼ Ckð Þð pR þ pRð Ukð Þþ Ukð Þ Ckð Þ Þþ gkð Þ ð Ukð Þ Ckð Þ ÞÞ:
As in the previous subsection, we may drop the last effective potential, ρ ¼ ρ0;k. In this case, one can also term in (32) due to consistency and thus the first two consider the broken phase expansion terms provide the result for the fully nonperturbative flow of the Yukawa coupling. One also notes that if the term in gkðρÞ¼g0;k þ g1;k · ðρ − ρ0;kÞþ ; ð33Þ question is not projected out and considered in, e.g., a ðs0;s8Þ (physical) background, then it cannot even be which, via (32) defines the broken phase flows of the g0;k interpreted as a standard Yukawa term, as in such back- and g1;k couplings. They can be found in Appendix C. † grounds ðM5M5 − ρ=3Þ ∼ T8. The presence of such con- tribution would force us to introduce in addition a different † IV. EFFECT OF THE M5M M5 OPERATOR Yukawa coupling and treat it similarly to as it was an 5 explicit symmetry breaking term. Motivated by the nonuniqueness of the definition of the The expression obtained for the flow of the Yukawa field-dependent Yukawa coupling in theR earlier setting, now ρ ¯ † coupling gkð Þ is thought to be a resummation of operators we investigate the case when the term x g53;kqM5M5M5q as described below (9). As such, one is able to determine is added to the ansatz (15) of the effective action (without how strong the interaction is between quarks and mesons subtracting ρqM¯ 5q=3 from it). Here g53;k is a new running as, e.g., the chiral condensate evaporates at high temper- coupling constant, and we will not be considering its field ature. In principle, gkðρÞ also depends explicitly on the dependence. Also, we restrict ourselves to calculations in temperature, but one expects that the decrease in ρ is more the symmetric phase; therefore, a similar treatment as of in important [5]. For phenomenology, one needs to evaluate Sec. III A is in order. All calculations presented there are gkðρÞ and its derivatives at the minimum point of the still valid, but there is one more term contributing to rhs of
056015-7 G. FEJŐS and A. PATKÓS PHYS. REV. D 103, 056015 (2021) the flow equation of the effective action, coming from the section, we wish to provide at least a convincing evidence tadpole diagram of Fig. 2 [see the mass matrices in (B9)], of how important it is to consider the field-dependent Z version of the Yukawa interaction. We will be investigating ¯ g53 ∂˜ ij † † the approximation scheme described in Sec. III C and q k½Gk;sðpRÞðfM5; fTi;Tjgg þ TiM5Tj þ TjM5TiÞ 2 p solve the flow equation for the Yukawa coupling when ij † † expanded around the minimum point of the potential. That G p M5; T ;T − T M T − T M T þ k;πð RÞðf f i jgg i 5 j j 5 iÞ is, we are dealing with (C1) and (C2) numerically. In ij γ † † this section, we employ Litim’s optimal regulator, þ Gk;sπðpRÞi 5ð½M5;½Ti;Tj þ TiM5Tj þ TjM5TiÞ R ðpÞ¼ðk2 − p2ÞΘðk2 − p2Þ; thus, p2 ¼ k2,ifp
∂ − 6 10 3 00 2 2 kg53;k ¼ ð g1;kCk þ g53;kCk þ g53;kUkÞ ∂ − Z Vch;k¼0 mK ffiffiffimπ 2 − 1 2 ¼ p sns þ mKss hs; ð42bÞ ˜ 3 00 3 ∂ss 2 × ∂ þ g0 U − g0 C k ðp2 þ U0 Þ2 ;k k 3 ;k k Zp R k 1 where the sns and ss condensates are defined analogously to ∂˜ their respective external fields. If we combine (42) with × k 2 2 0 2 : ð38Þ p pRðpR þ UkÞ (40), we arrive at the conclusion that irrespectively of the remaining model parameters (in particular, on including the We wish to note that based on Sec. III B, a broken phase axial anomaly), in the minimum point of the effective calculation can also be done. This, however, leads to such action complicated formulas that we do not go into the details. pffiffiffi s ¼ fπ;s¼ 2ðf − fπ=2Þ: ð43Þ V. NUMERICS ns;min s;min K Though a complete solution of the flow equation for the That is to say, in the minimum point, the ρ invariant ρ ≡ ρ 2 2 2 ≈ 93 5 2 effective action is beyond the scope of the paper, in this is 0 min ¼ðsns;min þ ss;minÞ= ð . MeVÞ .
056015-8 FIELD DEPENDENCE OF THE YUKAWA COUPLING IN THE … PHYS. REV. D 103, 056015 (2021)
TABLE I. Comparison between Yukawa couplings with and value with the one obtained at k ¼ 0 shows the importance without field dependence. Note that, in the former scenario, of considering a field-dependent Yukawa coupling. the coupling does not flow and is equal to its UV value. The flow equations were solved with a UV cutoff Λ ¼ 1 GeV and Δ − VI. SUMMARY g ¼ðg0;k¼0 g0;k¼ΛÞ=g0;k¼Λ. In this paper, we raised the question of determining the 00 Δ U g0;k¼Λ g0;k¼0 g field dependence of the Yukawa coupling in the three flavor 10 5 5.26 5.1% quark-meson model. Our main motivation was to clarify the 10 10 8.96 −10.4% problem of consistency between chiral symmetry and the 10 15 11.48 −23.4% Yukawa term. As opposed to two flavors, in case of three −30 4% 10 20 13.92 . flavors, the existence of non-Yukawa-like interactions and their generation in the infrared scales of the quantum 00 Δ U g0;k¼Λ g0;k¼0 g effective action prevents a straightforward generalization 20 5 5.24 4.7% of the two flavor approach [10]. That is to say, one cannot 20 10 8.97 −10.3% consistently work with, e.g., a scalar chiral condensate and 20 15 11.58 −22.8% associate the complete field dependence of the quark- 20 20 14.10 −29.5% quark-meson vertex with that of the Yukawa coupling itself. One needs great care to project out those operators that are not included in the ansatz of the effective action, Since we are interested in a rough estimate, (C1) and which, by construction, needs to respect chiral symmetry. (C2) will be solved such that the scale dependence of the We believe that such a consistent approach was missing in chiral potential is neglected, and we are only after g0;k (and the literature so far. 0 g1;k)atk ¼ . The expansion of the chiral potential around We have explicitly calculated the renormalization group the minimum reads flow of the field-dependent Yukawa coupling separately in the symmetric and broken phases. We have showed that at ρ 0 ρ ρ − ρ 00 ρ ρ − ρ 2 2 ρ τ Vchð Þ¼U ð 0Þð 0ÞþU ð 0Þð 0Þ = þ Cð 0Þ : the order of dimension 6 operators new type of terms arise and gave prescription for how to project them out as ð44Þ required by consistency. Motivated by the very definition of the flow of the field-dependent Yukawa coupling, for the The pion and kaon masses in the minimum are sake of a complete treatment of dimension 6 operators, we have determined the symmetric phase flows when these 2 0 C 2 2 mπ U s − 2s ; new (nonrenormalizable) interactions are also included in ¼ þ 6 ð ns;min s;minÞ ð45aÞ the ansatz of the quantum effective action (Γk). C pffiffiffi Numerics showed that the field dependence of the m2 U0 s2 − 3 2s s 4s2 ; Yukawa interaction is indeed important for physical para- K ¼ þ 6 ð ns;min ns;min s;min þ s;minÞ ð45bÞ metrizations of the model; therefore, it would be very 2 important to see what effects the current approach has from which using physical masses lead to U0 ≈ 0.147 GeV and a phenomenological point of view. One is typically C ≈ 84.37. We still need U00, which is determined via the interested in mapping the details of the chiral phase light scalar (σ) mass. Its expression reads transition at finite temperature and/or density, such as 1 the transition point, mass spectrum, interaction strengths, 2 0 00 2 2 mσ U U s s etc. To do so one also needs to include t’ Hooft’s ¼ þ 2 ð ns min þ s;minÞ determinant term into the system describing the U ð1Þ 1 pffiffiffiffi A C 2 10 2 − breaking, and it would be of particular interest to check the þ ðsns; min þ ss;minÞ D; 12 12 interplay between the anomaly coupling and that of the 5 6 00 2 4 4 7 3 00 2 4 D ¼ð C þ U Þ sns; min þ ð C þ U Þ ss;min Yukawa interaction. These directions represent future 4 19 2 105 00 − 18 002 2 2 studies to be reported elsewhere. þ ð C þ CU U Þss;minsns;min: ð46Þ
00 ACKNOWLEDGMENTS Setting 10 ≲ U ≲ 20 yields 469 MeV ≲ mσ ≲ 594 MeV, which is a suitable interval for the physical value of mσ. The authors thank Zs. Sz´ep for useful discussions For several different parameters, we show in Table I how related to the perturbative renormalization of the Yukawa the fluctuation corrected Yukawa coupling, g0;k¼0 differs coupling. This research was supported by the Hungarian from its UV value, g0;k¼Λ. Note that without field depend- National Research, Development and Innovation Fund ence, g0;k would not flow with respect to the scale at all and under Projects No. PD127982 and No. K104292. The stayed at its initial value. Therefore, comparing the initial work of G. F. was also supported by the János Bolyai
056015-9 G. FEJŐS and A. PATKÓS PHYS. REV. D 103, 056015 (2021)
Research Scholarship of the Hungarian Academy of diajdjbkdkcldldi Sciences and by the ÚNKP-20-5 New National 3 3 − Excellence Program of the Ministry for Innovation and ¼ 4 ðdabmdmcd þ dadmdmcbÞ 4 dacmdmbd Technology from the source of the National Research, 1 Development and Innovation Fund. δ δ δ δ δ δ þ 2 ð ab cd þ ac bd þ ad bcÞ rffiffiffi 1 3 δ 0d δ 0d δ 0d δ 0d ; APPENDIX A: U(3) ALGEBRA IDENTITIES þ 2 2ð a bcd þ b acd þ c abd þ d abcÞ
The Uð3Þ algebra is spanned by the 3 × 3 Ti ði ¼ 0; …8Þ ðA4aÞ generators, which satisfy TrðT T Þ¼δ =2, and a product i j ij f d d d þ f d d d ¼ f d d d ; of two generators lie in the Lie algebra, iaj jbk kcl ldi iaj jbk kdl lci abj ijk kcl ldi ðA4bÞ 1 f d d d þ f d d d ¼ f d d d TiTj ¼ 2 ðdijk þ ifijkÞTk: ðA1Þ iaj jck kdl lbi iaj jdk kcl lbi abj ijk kcl ldi þ 2fbdjflakfkjmdmcl þ 2fbcjfljkfkamdmdl: ðA4cÞ Associativity of matrix multiplication leads to the following identities: APPENDIX B: INVARIANT TENSORS AND TWO-POINT FUNCTIONS 0 ¼ filmfmjk þ fjlmfimk þ fklmfijm; ðA2aÞ First, we recall that
0 † ¼ filmdmjk þ fjlmdimk þ fklmdijm; ðA2bÞ ρ ¼ TrðM MÞ; ðB1aÞ
τ ¼ TrðM†M − TrðM†MÞ=3Þ2; ðB1bÞ 0 ¼ fijmfklm − dikmdjlm þ djlmdilm; ðA2cÞ
where M ¼ðsi þ iπiÞTi. In terms of si and πi, they read the first one known as the Jacobi identity. Using that TiTi is a Casimir operator, working in the adjoint representation, 1 ρ ¼ ðsisi þ πiπiÞ; ðB2aÞ one easily shows that fijkfljk ¼ 3δilð1 − δi0δl0Þ. Using this 2 identity as a starting point, one derives the following two- 1 τ π π π π and threefold sums using (A2): ¼ 24 ðsisjsksl þ i j k lÞDijkl þ s s π π ðD˜ − D =4Þ 3δ 3δ δ i j k l ij;kl ijkl dijmdkjm ¼ ik þ i0 k0; ðA3aÞ 1 − π π 2 12 ðsisi þ i iÞ ; ðB2bÞ 3 flnifikmfmjl ¼ − fnkj; ðA3bÞ 2 where Dijkl ¼ dijmdklm þ dikmdjlm þ dilmdjkm and ˜ Dij;kl ¼ dijmdklm. The relevant derivatives of ρ and τ are 3 dmikdknjfjlm ¼ finl; ðA3cÞ ∂ρ ∂ρ 2 ¼ si; ¼ πi; ðB3aÞ ∂si ∂πi rffiffiffi 3 ∂2ρ ∂2ρ ¼ δ ; ¼ δ ; ðB3bÞ finlfljmdmki ¼ ðδn0δjk þ δj0δnk − δk0δnjÞ; ij ij 2 ∂sisj ∂πiπj 3 − 2 dnjk ðA3dÞ ∂τ 2 1 ¼ − ρs þ s s s D rffiffiffi ∂s 3 i 6 a b c abci 3 i δ δ δ δ δ δ 1 dikldlnmdmji ¼ 2ð n0 jk þ j0 nk þ k0 njÞ π π 4 ˜ − þ 2 sa c dð Dai;cd DaicdÞ; ðB4aÞ 3 ∂τ 2 1 þ dnjk: ðA3eÞ 2 ¼ − ρπi þ πaπbπcDabci ∂πi 3 6 Furthermore, the following identities for fourfold sums 1 þ π s s ð4D˜ − D Þ; ðB4bÞ are useful for present calculations: 2 a c d ai;cd aicd
056015-10 FIELD DEPENDENCE OF THE YUKAWA COUPLING IN THE … PHYS. REV. D 103, 056015 (2021) 1 pffiffiffi ∂2τ 2 2 1 τ 2 8 2 − 4 2 2 js0;s8 ¼ s8ð s0 s0s8 þ s8Þ: ðB5bÞ ¼ − ρδij − sisj þ sasbDabij 24 ∂si∂sj 3 3 2 1 π π 4 ˜ − Then, the nonzero first derivatives are þ 2 c dð Dij;cd DijcdÞ; ðB4cÞ
∂2τ 2 2 1 ∂ρ ∂ρ ¼ − ρδ − π π þ π π D ∂π ∂π 3 ij 3 i j 2 a b abij ∂ ¼ s0; ∂ ¼ s8; ðB6aÞ i j s0 s0;s8 s8 s0;s8 1 4 ˜ − þ scsdð Dij;cd DijcdÞ; ðB4dÞ 2 ∂τ 2 2 s0 s8 ∂2τ 2 ¼ s8 − pffiffiffi ; ðB6bÞ ˜ ∂s0 3 3 2 ¼ − πisj þ saπcð4Daj;ci − DajciÞ: ðB4eÞ s0;s8 ∂πi∂sj 3 In case of the broken phase calculation, we are working ∂τ 2 2 2 s0 − s0sffiffiffi8 s8 in a background field M ¼ s0T0 þ s8T8; thus, the follow- ¼ s8 p þ : ðB6cÞ ∂s8 3 2 6 ing formulas need to be used. First, we list the invariants, s0;s8 1 2 2 As shown above, the second derivatives of ρ are equal to ρj ¼ ðs þ s Þ; ðB5aÞ s0;s8 2 0 8 the unit matrix, while that of τ are the following:
8 2 2 0 > 3 s8; if i ¼ j ¼ > > s2 4 > − p8ffiffi 0 8 8 0 > 2 þ 3 s0s8; if i ¼ ;j¼ or i ¼ ;j¼ > <> 2 pffiffiffi ∂2τ 2 2 s8 − 2 8 3 s0 þ 2 s0s8; if i ¼ j ¼ ¼ ðB7aÞ ∂ > 2 pffiffiffi sisj s0;s8 > 2 2 s8 > s þ þ 2s0s8; if i ¼ j ¼ 1; 2; 3 > 3 0 6 > 2 > 2 2 s8 1ffiffi > s0 þ − p s0s8; if i ¼ j ¼ 4; 5; 6; 7 :> 3 6 2 0; else 8 > 0; if i ¼ j ¼ 0 > > s2 2 > − p8ffiffi 0 8 8 0 > 3 2 þ 3 s0s8; if i ¼ ;j¼ or i ¼ ;j¼ > ffiffi <> 2 p ∂2τ s8 − 2 8 6 3 s0s8; if i ¼ j ¼ ¼ ffiffi : ðB7bÞ ∂π π > 2 p i j s0;s8 > s8 2 > − þ s0s8; if i ¼ j ¼ 1; 2; 3 > 6 3 > 5 2 1 ffiffi > s8 − p s0s8 if i ¼ j ¼ 4; 5; 6; 7 :> 6 3 2 0; else
The masses of the scalar and pseudoscalar mesons of the ansatz (15) in a purely bosonic background read
∂2V ∂2τ ðm2 Þ ¼ k ¼ δ ðU0 ðρÞþτC0 ðρÞÞ þ C ðρÞ s;k ij ∂s ∂s ij k k ∂s ∂s k i j i j ∂ρ ∂ρ ∂ρ ∂τ ∂ρ ∂τ 00 ρ τ 00 ρ 0 ρ þ ðUkð Þþ Ckð ÞÞ þ þ Ckð Þ; ðB8aÞ ∂si ∂sj ∂si ∂sj ∂sj ∂si
∂2V ∂2τ m2 k δ U0 ρ τC0 ρ C ρ ð π;kÞij ¼ ∂π ∂π ¼ ijð kð Þþ kð ÞÞ þ ∂π ∂π kð Þ i j i j ∂ρ ∂ρ ∂ρ ∂τ ∂ρ ∂τ 00 ρ τ 00 ρ 0 ρ þ ðUkð Þþ Ckð ÞÞ þ þ Ckð Þ; ðB8bÞ ∂πi ∂πj ∂πi ∂πj ∂πj ∂πi
056015-11 G. FEJŐS and A. PATKÓS PHYS. REV. D 103, 056015 (2021)
∂2V ∂2τ ∂ρ ∂ρ ∂ρ ∂τ ∂ρ ∂τ 2 k ρ 00 ρ τ 00 ρ 0 ρ ðmsπ;kÞij ¼ ¼ Ckð Þþ ðUkð Þþ Ckð ÞÞ þ þ Ckð Þ: ðB8cÞ ∂si∂πj ∂si∂sj ∂si ∂πj ∂si ∂πj ∂πj ∂si
If the fermions also have nonzero expectation values, then these masses get corrected by ∂ρ ∂M5 ∂ρ ∂M5 ∂ρ ∂ρ Δ 2 ¯ 0 ρ 0 ρ δ 00 ρ ðms;kÞij ¼ q gkð Þ þ þ gkð Þ ijM5 þ gkð Þ M5 q ∂si ∂sj ∂sj ∂si ∂si ∂sj † † þ g53q¯ðffTi;Tjg;M5gþTiM5Tj þ TjM5TiÞq; ðB9aÞ ∂ρ ∂M ∂ρ ∂M ∂ρ ∂ρ Δ 2 ¯ 0 ρ 5 5 0 ρ δ 00 ρ ðmπ;kÞij ¼ q gkð Þ þ þ gkð Þ ijM5 þ gkð Þ M5 q ∂πi ∂πj ∂πj ∂πi ∂πi ∂πj † † þ g53q¯ðfM5; fTi;Tjgg − TiM5Tj − TjM5TiÞq; ðB9bÞ ∂ρ ∂M5 ∂ρ ∂M5 ∂ρ ∂ρ Δ 2 ¯ 0 ρ 00 ρ ðmsπ;kÞij ¼ q gkð Þ þ þ gkð Þ M5 q ∂si ∂πj ∂πj ∂si ∂si ∂πj † † þ ig53q¯γ5ð½M5; ½Ti;Tj þ TiM5Tj þ TjM5TiÞq; ðB9cÞ
† where we also included the effect of the ∼qM¯ 5M5M5q operator with the coupling constant g53. Using the formulas above, in a purely bosonic background, which is defined by M ¼ s0T0 þ s8T8, the two-point functions of the ansatz (15) read 1 pffiffiffi ð2Þ00 2 0 0 2 00 4 2 0 00 2 3 0 00 2 4 2 00 Γ p p U 8 2C 5C s C s s − 4 2s0 3C C s s C C s s 24s U ; k;s ð Þ¼ þ k þ 24 ð ð k þ k 0 þ k 0Þ 8 ð k þ k 0Þ 8 þð k þ k 0Þ 8 þ 0 kÞ ðB10aÞ 1 pffiffiffi pffiffiffi ð2Þ08 0 2 0 00 2 2 0 00 2 3 00 4 Γ p s8 −12 2 C C s s8 4s0 5C 2C s s − 4 2 C C s s C s0s k;s ð Þ¼24 ð ð k þ k 0Þ þ ð k þ k 0Þ 8 ð k þ k 0Þ 8 þ k 8 8 4 2 0 2 3 00 þ s0ð Ck þ Cks0 þ UkÞÞ; ðB10bÞ 1 pffiffiffi ð2Þ88 2 0 2 0 2 00 4 0 2 00 4 Γ p p U 8s 2C 5C s C s − 4 2s0s8 6C 7C s C s k;s ð Þ¼ þ k þ 24 ð 0ð k þ k 8 þ k 8Þ ð k þ k 8 þ k 8Þ 2 12 9 0 2 00 4 24 00 þ s8ð Ck þ Cks8 þ Cks8 þ UkÞÞ; ðB10cÞ 1 pffiffiffi ð2Þ00 2 0 0 2 2 2 Γ p p U C s 8s − 4 2s0s8 s ; k;π ð Þ¼ þ k þ 24 k 8ð 0 þ 8Þ ðB10dÞ ffiffiffi ð2Þ08 1 p Γ p − C s8 −4s0 2s8 ; k;π ð Þ¼ 6 k ð þ Þ ðB10eÞ 1 pffiffiffi pffiffiffi ð2Þ88 2 0 0 2 2 Γ p p U s8 4C −2 2s0 s8 C s8 8s − 4 2s0s8 s ; k;π ð Þ¼ þ k þ 24 ð kð þ Þþ k ð 0 þ 8ÞÞ ðB10fÞ and furthermore, 1 pffiffiffi ð2Þ11 2 0 2 0 4 2 0 2 0 2 Γ p p U 4C s C s 8s 2C C s − 2 2s0s8 −12C 2C s ; k;s ð Þ¼ þ k þ 24 ð k 8 þ k 8 þ 0ð k þ k 8Þ ð k þ k 8ÞÞ ðB11aÞ 1 pffiffiffi ð2Þ44 2 0 2 0 4 2 0 2 0 2 Γ p p U 4C s C s 8s 2C C s − 2 2s0s8 6C 2C s ; k;s ð Þ¼ þ k þ 24 ð k 8 þ k 8 þ 0ð k þ k 8Þ ð k þ k 8ÞÞ ðB11bÞ 1 pffiffiffi pffiffiffi ð2Þ11 2 0 0 2 2 Γ p p U s8 4C 2 2s0 − s8 C s8 8s − 4 2s0s8 s ; k;π ð Þ¼ þ k þ 24 ð kð Þþ k ð 0 þ 8ÞÞ ðB11cÞ 1 pffiffiffi pffiffiffi ð2Þ44 2 0 0 2 2 Γ p p U s8 4C − 2s0 5s8 C s8 8s − 4 2s0s8 s ; k;π ð Þ¼ þ k þ 24 ð kð þ Þþ k ð 0 þ 8ÞÞ ðB11dÞ
056015-12 FIELD DEPENDENCE OF THE YUKAWA COUPLING IN THE … PHYS. REV. D 103, 056015 (2021)
Γð2Þ11 Γð2Þ22 Γð2Þ33 Γð2Þ44 Γð2Þ55 Γð2Þ66 Γð2Þ77 where k;s=π ðpÞ¼ k;s=π ðpÞ¼ k;s=π ðpÞ, and k;s=π ðpÞ¼ k;s=π ðpÞ¼ k;s=π ðpÞ¼ k;s=π ðpÞ.
APPENDIX C: BROKEN PHASE FLOWS OF g0 AND g1 Using the expansion (33) and the flow equation (32), we get Z 3 3 8 1 ∂ ∂˜ kg0;k ¼ k g1;k 2 0 þ 2 0 þ 2 0 00 p 2 p þ U 3p þ 4C ρ0 þ 3U p þ U þ 2ρ0U R k R k k R k k 3 2 2 2 0 00 2 0 g0 ð3p − g0 ρ0Þð16Ckðp þ U Þþ3U ðp þ 12Ckρ0 þ U ÞÞρ0 þ ;k R ;k R k k R k ; ðC1Þ 3 2 2 ρ 2 2 0 3 2 4 ρ 3 0 2 0 2ρ 00 ð pR þ g0;k 0Þ ðpR þ UkÞð pR þ Ck 0 þ UkÞðpR þ Uk þ 0UkÞ Z 3 2 2 2 0 00 2 0 g ð3p − g ρ0Þð16C ðp þ U Þþ3U ðp þ 12C ρ0 þ U ÞÞ ∂ ∂˜ 0;k R 0;k k R k k R k k kg1;k ¼ k 2 2 2 2 0 2 0 2 0 00 ð3p þ g ρ0Þ ðp þ U Þð3p þ 4C ρ0 þ 3U Þðp þ U þ 2ρ0U Þ p R 0;k R k R k k R k k
3 ∂ 3 8 1 þ g1;k 2 0 þ 2 0 þ 2 0 00 2 ∂ρ p þ U ðρÞ 3p þ 4C ðρÞρ þ 3U ðρÞ p þ U ðρÞþ2ρU ðρÞ ρ R k R k k R k k 0 ∂ 3 ρ 3 2 − 2 ρ ρ 16 ρ 2 0 ρ 3 00 ρ 2 12 ρ ρ 0 ρ gkð Þð pR gkð Þ Þð Ckð ÞðpR þ Ukð ÞÞ þ Ukð ÞðpR þ Ckð Þ þ Ukð ÞÞÞ þ ρ0 ; ðC2Þ ∂ρ 3 2 2 ρ ρ 2 2 0 ρ 3 2 4 ρ ρ 3 0 ρ 2 0 ρ 2ρ 00 ρ ð pR þ gkð Þ Þ ðpR þ Ukð ÞÞð pR þ Ckð Þ þ Ukð ÞÞðpR þ Ukð Þþ Ukð ÞÞ ρ0 where gkðρÞ¼g0;k þ g1;kðρ − ρ0Þ, and it is understood that unless indicated otherwise, all couplings are taken at the minimum point, ρ0.
[1] J. Berges, N. Tetradis, and C. Wetterich, Phys. Rep. 363, [15] K. Otto, M. Oertel, and B.-J. Schaefer, Eur. Phys. J. Special 223 (2002). Topics 229, 3629 (2020). [2] H. Gies, Lecture Notes in Physics (Springer, Berlin, [16] A. K. Cyrol, M. Mitter, J. M. Pawlowski, and N. Strodthoff, Heidelberg, 2012), Vol. 852. Phys. Rev. D 97, 054006 (2018). [3]N.Dupuis,L.Canet,A.Eichhorn,W.Metzner, [17] R. Alkofer, A. Maas, W. A. Mian, M. Mitter, J. Paris-Lopez, J.M.Pawlowski,M.Tissier,andN.Wschebor,Phys. J. M. Pawlowski, and N. Wink, Phys. Rev. D 99, 054029 Rep. https://doi.org/10.1016/j.physrep.2021.01.001 (to be (2019). published). [18] F. Rennecke and B.–J. Schaefer, Phys. Rev. D 96, 016009 [4] G. v. Gersdorff and C. Wetterich, Phys. Rev. B 64, 054513 (2017). (2001). [19] D.-U. Jungnickel and C. Wetterich, Phys. Rev. D 53, 5142 [5] G. Fejos and A. Hosaka, Phys. Rev. D 98, 036009 (2018). (1996). [6] M. E. Peskin and D. V. Schroeder, An Introduction to Quan- [20] A. Patkos, Mod. Phys. Lett. D 27, 1250212 (2012). tum Field Theory (Westview Press, Boulder, CO, 1995). [21] M. Grahl and D. H. Rischke, Phys. Rev. D 88, 056014 [7] H. Gies and C. Wetterich, Phys. Rev. D 69, 025001 (2004). (2013). [8] J. Braun, Eur. Phys. J. C 64, 459 (2009). [22] M. Grahl, Phys. Rev. D 90, 117904 (2014). [9] J. Berges, D.-U. Jungnickel, and C. Wetterich, Phys. Rev. D [23] G. Fejos, Phys. Rev. D 90, 096011 (2014). 59, 034010 (1999). [24] H. Gies, R. Sondenheimer, and M. Warschinke, Eur. Phys. [10] J. M. Pawlowski and F. Rennecke, Phys. Rev. D 90, 076002 J. C 77, 743 (2017). (2014). [25] G.-P. Vacca and L. Zambelli, Phys. Rev. D 91, 125003 [11] M. Mitter and B.-J. Schaefer, Phys. Rev. D 89, 054027 (2015). (2014). [26] C. Vafa and E. Witten, Nucl. Phys. B234, 173 (1984). [12] K. Kamikado and T. Kanazawa, J. High Energy Phys. 01 [27] C. Wetterich, Phys. Lett. B 301, 90 (1993). (2015) 129. [28] T. R. Morris, Int. J. Mod. Phys. A 09, 2411 (1994). [13] Y. Jiang, T. Xia, and P. Zhuang, Phys. Rev. D 93, 074006 [29] K.-I. Aoki, K. I. Morikawa, J. I. Sumi, H. Terao, and M. (2016). Tomoyose, Prog. Theor. Phys. 97, 479 (1997). [14] S. Resch, F. Rennecke, and B.-J. Schaefer, Phys. Rev. D 99, [30] A. Jakovac and A. Patkos, Phys. Rev. D 88, 065008 076005 (2019). (2013).
056015-13