Heavy-quark spin polarization induced by the Kondo effect in a magnetic field

We propose a new mechanism of the heavy-quark spin polarization (HQSP) in quark matter induced by the Kondo effect under external magnetic field. The Kondo effect is caused by a condensate between a heavy and a light quark called the Kondo condensate leading to a mixing of the heavy and light quark spins. Thus the HQSP is driven through the Kondo effect from light quarks coupling with the magnetic field in quark matter. For demonstration, we employ the Nambu--Jona-Lasinio type model under a magnetic field, and investigate the HQSP within the linear response theory with vertex corrections required by the $U(1)_{\rm EM}$ electromagnetic gauge invariance. As a result, we find that the HQSP arises significantly with the appearance of the Kondo effect. Our findings are testable in future sign-problem-free lattice simulations.


I. INTRODUCTION
The Kondo effect is known as one of the important phenomena in the quantum many-body system [1,2].This effect arises when heavy impurities exist in matter coupling with itinerant fermions through non-Abelian interactions such as the spin exchange.The Kondo effect changes transport properties significantly; e.g., it turns the electrical resistivity from decrement to increment with the temperature lowered.
Recently, it has been discussed that the Kondo effect persists universally in high-energy physics as well, although the effect was originally discovered in condensedmatter physics.For example, in the context of quantum chromodynamics (QCD), the effect is driven when heavy quarks (c, b) exist as an impurity in dense quark matter formed by light quarks (u, d) [3,4].This is particularly referred to as the QCD Kondo effect.In this case, the non-Abelian interaction is supplied by SU (N c ) color exchange (N c = 3 is the number of colors).The Kondo effect is expected to arise also in the hadronic phase at lower density.In fact, it was shown that the Kondo effect arises in nuclear matter when Σ c and Σ * c baryons or D and D * mesons exist as impurities, where the spin and/or isospin exchange serves as the non-Abelian interaction [3,[5][6][7].In addition, many works on the Kondo effect in a relativistic system have been done in the literatures [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].In this way, the interdisciplinary study bridging condensed-matter physics and QCD/hadron physics focused on the Kondo effect is at-tracting attention.
In a field-theoretical treatment, the Kondo effect is caused by a condensate (a hybridization) formed by a light itinerant fermion and a heavy impurity called Kondo condensate.This condensate not only provides a gap of the itinerant fermions near the Fermi level but also causes a mixing between the itinerant fermion and the heavy impurity as induced by an s-d interaction in the Anderson model [27].In quark matter, due to the mixing, in Ref. [25], the magnetically induced axial current of light quarks in quark matter, namely, the chiral separation effect (CSE) [28][29][30], was found to be enhanced under the Kondo effect.In solid states, in Ref. [23], the magnetic responses of a Dirac and nonrelativistic bands with their hybridizations were investigated, showing that the spin-orbit crossed part of the magnetic susceptibility of the Dirac (nonrelativistic) band is enhanced (suppressed) due to the presence of the hybridizations.Those findings show that the Kondo condensate (hybridization) has a great impact on magnetic responses of the fermions.
In this paper, we propose a new mechanism of the heavy-quark spin polarization (HQSP) induced by the Kondo effect under a magnetic field.When the Kondo effect is absent, the HQSP does not arise significantly, since the Zeeman interaction of a heavy quark is of order 1/m Q (m Q is the mass of a heavy quark) and is suppressed for sufficiently large m Q .In particular, it vanishes at m Q → ∞.On the other hand, when the Kondo effect is present, the HQSP can arise without 1/m Q suppression.The Kondo condensate correlates the heavy-quark spin with the light-quark spin coupling to the magnetic field in a medium.In other words, the condensate converts the magnetic response of the light-quark spin into that of heavy-quark spin.Schematically this effect is depicted in Fig. 1.This new mechanism of the HQSP is expected to be testable in future sign-problem-free lattice simula-tions.
This paper is organized as follows.In Sec.II, we introduce the model for demonstration and derive the Green's function of the fermions.In Sec.III, we show the analytic evaluation for the HQSP within the linear response theory with the vertex corrections.In Sec.IV, we study the HQSP response function in the so-called dynamical and static limits in detail, toward a better understanding of the HQSP in the timelike and spacelike momentum regions.Referring to it, in Sec.V, we present numerical results of the HQSP in both the momentum regions.In Sec.VI, we compare the HQSP induced by the Kondo effect with that by the ordinary Zeeman effect, and in Sec.VII, we conclude the present work.

II. MODEL
Here, we present our effective model to study the HQSP induced by the Kondo effect under a magnetic field.One of the most useful models for the demonstration is the Nambu-Jona-Lasinio (NJL) type model including a four-point interaction between a light and a heavy quark [9,13].Namely, we start with the following Lagrangian: In this Lagrangian, ψ is the light quark described by the ordinary Dirac theory, while Ψ v is the heavy one described by the heavy-quark effective theory (HQET) [31][32][33][34].Thus, Ψ v is defined by Ψ v = e im Q t 1+γ0 2 Ψ, with Ψ being the relativistic heavy fermion in Dirac theory, and µ is the chemical potential of the light quark included to access finite density.It should be noted that λ is a Lagrange multiplier included to impose a condition Ψ † v Ψ v = n Q , with n Q a space-averaged heavy-quark density.In the following analysis, we examine the HQSP for the λ = 0 case as a clear demonstration.The common coupling g for the scalar and vector interactions is derived by the Fierz transformation from the one-gluon exchange interaction [35].In Eq. ( 1) the covariant derivatives are defined by where the electromagnetic gauge field A µ = (A 0 , A) is introduced to incorporate interactions with the magnetic field.The coupling constants e q and e Q are electric charges of the light and heavy quarks, respectively.We take e q = +2 3 e for the u quark, e q = −1 3 e for the d quark, and e Q = + 2  3 e for the c quark, with the elementary charge e.
For the Lagrangian (1), we introduce the Kondo condensate made of the light and heavy quarks to describe the ground state governed by the Kondo effect.For this purpose, we make use of the mean field approximation for ψΨ v and ψγ i Ψ v by rewriting the Lagrangian (1) as with λ = 0 taken.In the previous works [9,13], it was found that the reasonable ground state is realized by assuming the so-called hedgehog ansatz provided by with pi ≡ p i /|p| defined in momentum space, where p is identical to the momentum of the heavy and light quarks. 1In Eq. ( 4), ∆ serves as the Kondo condensate made of a light quark and a heavy quark.The value of ∆ should be determined by a variational method on the free energy of the system.Despite the violation of O(3) rotational invariance due to the magnetic field, we employ the ansatz (4) as an appropriate configuration of the Kondo effect since we make use of the linear response theory for a weak magnetic field.The nontrivial phase characterized by ∆ = 0 is called the Kondo phase, whereas the trivial phase by ∆ = 0 is called the normal phase. 2  From the Lagrangian (3) together with the ansatz (4), the inverse of Green's function of the fermions in the absence of a gauge field is read as This Green's function is a 6×6 matrix since the light and heavy quarks are four-component and two-component spinors, respectively.The inverse of Eq. ( 5) yields the Green's function which is of the form [25] G0 (p 0 , p) = G ψψ 0 (p 0 , p) with the elements The matrices Λ p and Λ a are the projection operators for positive-and negative-energy states of the light quark defined by with α = γ 0 γ, respectively, and 1 in Eq. ( 7) is a 2×2 unit matrix.Similarly, Λ pH and Λ Hp are the operators mixing the positive-energy states of light and heavy quarks defined by respectively.The dispersion relations E + p , E − p , and E a p are given by In the following analysis, the modes carrying the dispersions E + p , E − p , and E a p are referred to as the Kondo quasiparticle ("q.p."), the Kondo quasihole ("q.h."), and the light antiparticle ("a.p."), respectively.The factors 0.0 0.2 0.4 0.6 0.8 1.0 -1.0 -0.5 0.0 0.5 1.0 energy p 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " T u F D f H N p p e P Y / u i 0 S C l 8 P 2 V z Z v U = " > A A A C d n i c h V G 7 S g N B F D 1 Z 3 / G V a C M I E g x R q 3 B X B M V K t L E 0 i Y m C k b C 7 j n F x X + x u I j H 4 A / 6 A h V g o a B A / w 8 Y f s M g n i G U E L S y 8 2 S y I i n q H m T l z 5 p 4 7 Z 2 Z U x 9 A 9 n 6 g Z k b q 6 e 3 r 7 + g e i g 0 P D I 6 O x + F j B s y u u J v K a b d j u t q p 4 w t A t k f d 1 3 x D b j i s U U z X E l n q 4 1 t 7 f q g r X 0 2 1 r 0 6 8 5 Y t d U y p a + r 2 u K z 1 Q p F q 8 X X T M h L O G W a y f F h F O i U i x J a Q o i 8 R P I I U g i j A 0 7 1 k A R e 7 C h o Q I T A h Z 8 x g Y U e N x 2 I I P g M L e L O n M u I z 3 Y F z h B l L U V z h K c o T B 7 y G O Z V z s h a / G 6 X d M L 1 B q f Y n B 3 W Z l A i h 7 p l l r 0 Q H f 0 R O + / 1 q o H N d p e a j y r H a 1 w S q O n E 7 n X f 1 U m z z 4 O P l V / e v a x j 6 X A q 8 7 e n Y B p 3 0 L r 6 K v H Z 6 3 c c j Z V n 6 E r e m b / l 9 S k e 7 6 B V X 3 R r j M i e 4 4 o f 4 D 8 / b l / g s J 8 W q a 0 n F l I r q y G X 9 G P S U x j j t 9 7 E S t Y x w b y f O 4 R L n C D R u R N m p J S 0 m w n V Y q E m n F 8 C Y k + A M p U k K k = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " T u F D f H N p p e P Y / u i 0 S C l 8 P 2 V z Z v U = " > A A A C d n i c h V G 7 S g N B F D 1 Z 3 / G V a C M I E g x R q 3 B X B M V K t L E 0 i Y m C k b C 7 j n F x X + x u I j H 4 A / 6 A h V g o a B A / w 8 Y f s M g n i G U E L S y 8 2 S y I i n q H m T l z 5 p 4 7 Z 2 Z U x 9 A 9 n 6 g Z k b q 6 e 3 r 7 + g e i g 0 P D I 6 O x + F j B s y u u J v K a b d j u t q p 4 w t A t k f d 1 3 x D b j i s U U z X E l n q 4 1 t 7 f q g r X 0 2 1 r 0 6 8 5 Y t d U y p a + r 2 u K z e a j y r H a 1 w S q O n E 7 n X f 1 U m z z 4 O P l V / e v a x j 6 X A q 8 7 e n Y B p 3 0 L r 6 K v H Z 6 3 c c j Z V n 6 E r e m b / l 9 S k e 7 6 B V X 3 R r j M i e 4 4 o f 4 D 8 / b l / g s J 8 W q a 0 n F l I r q y G X 9 G P S U x j j t 9 y 8 2 S y I i n q H m T l z 5 p 4 7 Z 2 Z U x 9 A 9 n 6 g Z k b q 6 e 3 r 7 + g e i g 0 P D I 6 O x + F j B s y u u J v K a b d j u t q p 4 w t A t k f d 1 3 x D b j i s U U z X E l n q 4 1 t 7 f q g r X 0 2 1 r 0 6 8 5 Y t d U y p a + r 2 u K z e a j y r H a 1 w S q O n E 7 n X f 1 U m z z 4 O P l V / e v a x j 6 X A q 8 7 e n Y B p 3 0 L r 6 K v H Z 6 3 c c j Z V n 6 E r e m b / l 9 S k e 7 6 B V X 3 R r j M i e 4 4 o f 4 D 8 / b l / g s J 8 W q a 0 n F l I r q y G X 9 G P S U x j j t 9 y 8 2 S y I i n q H m T l z 5 p 4 7 Z 2 Z U x 9 A 9 n 6 g Z k b q 6 e 3 r 7 + g e i g 0 P D I 6 O x + F j B s y u u J v K a b d j u t q p 4 w t A t k f d 1 3 x D b j i s U U z X E l n q 4 1 t 7 f q g r X 0 2 1 r 0 6 8 5 Y t d U y p a + r 2 u K z e a j y r H a 1 w S q O n E 7 n X f 1 U m z z 4 O P l V / e v a x j 6 X A q 8 7 e n Y B p 3 0 L r 6 K v H Z 6 3 c c j Z V n 6 E r e m b / l 9 S k e 7 6 B V X 3 R r j M i e 4 4 o f 4 D 8 / b l / g s J 8 W q a 0 n F l I r q y G X 9 G P S U x j j t 9 X a H t p 8 K x 2 t M L e i x 5 P 5 d 7 / V V V 5 9 n D w p f r T s 4 c S l n 2 v O n u 3 f a Z 9 C 6 2 j r x + e t 3 I r 2 W R z j q 7 o l f 1 f 0 h P d 8 w 3 M + p t 2 n R H Z C 0 T 4 A + S f z 9 0 N t h Z S M q X k z G I i v R p 8 R R j T m M U 8 v / c S 0 l j H B v J 8 r s A J T n E W e p a G p Q l p s p M q h Q L N B L 6 F N P M J Z k y J s A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " H + m e I Y q 2 I 5 S q a o i C W l l r 7 x f q w n F 1 y 9 z 0 G r b Y r S p l U y / p m u I x l a G 9 W I J S 5 E e 8 G 8 g B S C C I D S t 2 g x 3 s w X a H t p 8 K x 2 t M L e i x 5 P 5 d 7 / V V V 5 9 n D w p f r T s 4 c S l n 2 v O n u 3 f a Z 9 C 6 2 j r x + e t 3 I r 2 W R z j q 7 o l f 1 f 0 h P d 8 w 3 M + p t 2 n R H Z C 0 T 4 A + S f z 9 0 N t h Z S M q X k z G I i v R p 8 R R j T m M U 8 v / c S 0 l j H B v J 8 r s A J T n E W e p a G p Q l p s p M q h Q L N B L 6 F N P M J Z k y J s A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " p r l 2 k K P l e 4 B 3 f 9 w d M Y B Z z q P K + h / i M L / i q 6 d q w N q a N t 0 q 1 t k L T i 9 9 C m / 0 J z o O Y 8 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " p r l 2 k K P l e 4 B 3 f 9 w d M Y B Z z q P K + h / i M L / i q 6 d q w N q a N t 0 q 1 t k L T i 9 9 C m / 0 J z o O Y 8 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " p r l 2 k K P l e 4 B 3 f 9 w d M Y B Z z q P K + h / i M L / i q 6 d q w N q a N t 0 q 1 t k L T i 9 9 C m / 0 J z o O Y 8 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " b s 8 x q Z l z b P T l 8 r o v r o 5 2 c r l / e q f J 4 V d v 6 q 7 v S s s I 3 p 3 K t k 7 1 H O Z L d w + / r W 3 t f z l Z n l q f Q F H d A f 9 r 9 P P f r B N w h a F + 7 h k l j + h j I 3 w P z / u W + C 1 W r F p I q 5 R C h h A s / x k p / 5 D e b x H o u o 8 X F f c I o z f N d 0 7 b U 2 2 2 + V N l D 0 b B z / h D Z 3 D T K + l / g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " x q Z l z b P T l 8 r o v r o 5 2 c r l / e q f J 4 V d v 6 q 7 v S s s I 3 p 3 K t k 7 1 H O Z L d w + / r W 3 t f z l Z n l q f Q F H d A f 9 r 9 P P f r B N w h a F + 7 h k l j + h j I 3 w P z / u W + C 1 W r F p I q 5 R C h h A s / x k p / 5 D e b x H o u o 8 X F f c I o z f N d 0 7 b U 2 2 2 + V N l D 0 b B z / h D Z 3 D T K + l / g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " U A V + q z u 9 z O e i a 5 2 u V d P P 5 j V h o Y p Q v 6 T t d 0 T j / o J 9 3 9 t V e a 9 8 i 8 7 P L s d L Q i a g x 8 e r l 4 + 1 + V z 7 P C x 3 v V P z 0 r b K K a e 5 X s P c q Z 7 B R u R 9 / a + 3 K 9 O L k w m r 6 i Q 7 p i / w d 0 S W d 8 g q B 1 4 x 7 N i 4 W v K P E D m H 9 e 9 0 O w P F 4 x q W L O U 3 l 6 p n i K P g z q h n 3 T 9 z 1 x J m q P v p c O z P d A K a U 6 9 n 9 m 8 + q / K 5 1 n h 4 F Z 1 r 2 e F J k q p V 5 e 9 y 5 T p v 8 I Z 6 N v v P l 5 u l j f m k w X 6 T L / Y / y f q 0 R m / I G j / d r 6 s i 4 z q h n 3 T 9 z 1 x J m q P v p c O z P d A K a U 6 9 n 9 m 8 + q / K 5 1 n h 4 F Z 1 r 2 e F J k q p V 5 e 9 y 5 T p v 8 I Z 6 N v v P l 5 u l j f m k w X 6 T L / Y / y f q 0 R m / I G j / d r 6 s i 4 x q n a 4 V M 8 7 h E r C + i n S z q l W 7 q g M 7 q m 3 / + s l a Q 1 6 l 7 2 e L Y b W i H N 7 o P e h f v / q n y e F b Y f V c 9 6 V t j E S O r V Z e 8 y Z e q 3 c B r 6    x F q + D m l 6 o 1 v g U g 7 v L y j R m 6 I F u 6 I X u 6 Z a e 6 P 3 X W s 2 w R u C l w b P a 0 Q p n L 3 k y W X r 7 V 2 X y 7 O P w U / W n Z x 8 H W A q 9 6 u z d C Z n g F l p H X z 9 q v Z S W i z P N W b q k Z / Z / Q W 2 6 4 x t Y 9 V f t q i C K 5 0 j w B 8 j f n / s n 2 J j P y p S V C w u Z 3 E r 0 F X F M Y R p z / N 6 L y G E N e Z T 5 3 A p O c Y Z W 7 F E a l s a l i U 6 q F I s 0 Y / g S U v o D z l G K 0 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " t q G 6 F h A X x F q + D m l 6 o 1 v g U g 7 v L y j R m 6 I F u 6 I X u 6 Z a e 6 P 3 X W s 2 w R u C l w b P a 0 Q p n L 3 k y W X r 7 V 2 X y 7 O P w U / W n Z x 8 H W A q 9 6 u z d C Z n g F l p H X z 9 q v Z S W i z P N W b q k Z / Z / Q W 2 6 4 x t Y 9 V f t q i C K 5 0 j w B 8 j f n / s n 2 J j P y p S V C w u Z 3 E r 0 F X F M Y R p z / N 6 L y G E N e Z T 5 3 A p O c Y Z W 7 F E a l s a l i U 6 q F I s 0 Y / g S U v o D z l G K 0 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " t q G 6 F h A X x F q + D m l 6 o 1 v g U g 7 v L y j R m 6 I F u 6 I X u 6 Z a e 6 P 3 X W s 2 w R u C l w b P a 0 Q p n L 3 k y W X r 7 V 2 X y 7 O P w U / W n Z x 8 H W A q 9 6 u z d C Z n g F l p H X z 9 q v Z S W i z P N W b q k Z / Z / Q W 2 6 4 x t Y 9 V f t q i C K 5 0 j w B 8 j f n / s n 2 J j P y p S V C w u Z 3 E r 0 F X F M Y R p z / N 6 L y G E N e Z T 5 3 A p O c Y Z W 7 F E a l s a l i U 6 q F I s 0 Y / g S U v o D z l G K 0 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " b s 8 x F q + D m l 6 o 1 v g U g 7 v L y j R m 6 I F u 6 I X u 6 Z a e 6 P 3 X W s 2 w R u C l w b P a 0 Q p n L 3 k y W X r 7 V 2 X y 7 O P w U / W n Z x 8 H W A q 9 6 u z d C Z n g F l p H X z 9 q v Z S W i z P N W b q k Z / Z / Q W 2 6 4 x t Y 9 V f t q i C K 5 0 j w B 8 j f n / s n 2 J j P y p S V C w u Z 3 E r 0 F X F M Y R p z / N 6 L y G E N e Z T 5 3 A p O c Y Z W 7 F E a l s a l i U 6 q F I s 0 Y / g S U v o D z l G K 0 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " t q G x F q + D m l 6 o 1 v g U g 7 v L y j R m 6 I F u 6 I X u 6 Z a e 6 P 3 X W s 2 w R u C l w b P a 0 Q p n L 3 k y W X r 7 V 2 X y 7 O P w U / W n Z x 8 H W A q 9 6 u z d C Z n g F l p H X z 9 q v Z S W i z P N W b q k Z / Z / Q W 2 6 4 x t Y 9 V f t q i C K 5 0 j w B 8 j f n / s n 2 J j P y p S V C w u Z 3 E r 0 F X F M Y R p z / N 6 L y G E N e Z T 5 3 A p O c Y Z W 7 F E a l s a l i U 6 q F I s 0 Y / g S U v o D z l G K 0 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " t q G x F q + D m l 6 o 1 v g U g 7 v L y j R m 6 I F u 6 I X u 6 Z a e 6 P 3 X W s 2 w R u C l w b P a 0 Q p n L 3 k y W X r 7 V 2 X y 7 O P w U / W n Z x 8 H W A q 9 6 u z d C Z n g F l p H X z 9 q v Z S W i z P N W b q k Z / Z / Q W 2 6 4 x t Y 9 V f t q i C K 5 0 j w B 8 j f n / s n 2 J j P y p S V C w u Z 3 E r 0 F X F M Y R p z / N 6 L y G E N e Z T 5 3 A p O c Y Z W 7 F E a l s a l i U 6 q F I s 0 Y / g S U v o D z l G K 0 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " t q G x F q + D m l 6 o 1 v g U g 7 v L y j R m 6 I F u 6 I X u 6 Z a e 6 P 3 X W s 2 w R u C l w b P a 0 Q p n L 3 k y W X r 7 V 2 X y 7 O P w U / W n Z x 8 H W A q 9 6 u z d C Z n g F l p H X z 9 q v Z S W i z P N W b q k Z / Z / Q W 2 6 4 x t Y 9 V f t q i C K 5 0 j w B 8 j f n / s n 2 J j P y p S V C w u Z 3 E r 0 F X F M Y R p z / N 6 L y G E N e Z T 5 3 A p O c Y Z W 7 F E a l s a l i U 6 q F I s 0 Y / g S U v o D z l G K 0 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " t q G x F q + D m l 6 o 1 v g U g 7 v L y j R m 6 I F u 6 I X u 6 Z a e 6 P 3 X W s 2 w R u C l w b P a 0 Q p n L 3 k y W X r 7 V 2 X y 7 O P w U / W n Z x 8 H W A q 9 6 u z d C Z n g F l p H X z 9 q v Z S W i z P N W b q k Z / Z / Q W 2 6 4 x t Y 9 V f t q i C K 5 0 j w B 8 j f n / s n 2 J j P y p S V C w u Z 3 E r 0 F X F M Y R p z / N 6 L y G E N e Z T 5 3 A p O c Y Z W 7 F E a l s a l i U 6 q F I s 0 Y / g S U v o D z l G K 0 g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " t q G x F q + D m l 6 o 1 v g U g 7 v L y j R m 6 I F u 6 I X u 6 Z a e 6 P 3 X W s 2 w R u C l w b P a 0 Q p n L 3 k y W X r 7 V 2 X y 7 O P w U / W n Z x 8 H W A q 9 6 u z d C Z n g F l p H X z 9 q v Z S W i z P N W b q k Z / Z / Q W 2 6 4 x t Y 9 V f t q i C K 5 0 j w B 8 j f n / s n 2 J j P y p S V C w u Z 3 E r 0 F X F M Y R p z / N 6 L y G E N e Z T 5 3 A p O c Y Z W 7 F E a l s a l i U 6 q F I s 0 Y / g S U v o D z l G K 0 g = = < / l a t e x i t >

Fermi level
< l a t e x i t s h a 1 _ b a s e 6 4 = " I f S i 8 9 9 E v e Z J i X h y J F e 7 f I r P P W a l j l E 6 o y O 6 p B P 6 T j / p 5 q + 1 0 r x G 2 8 s u z 0 5 H K 6 Q 1 s P 9 8 8 f q / q o B n h e 1 f q n 9 6 V t j E Z O 7 V Y + 8 y Z 9 q 3 c D v 6 1 t 6 n y 8 W p 2 m j 6 g r 7 R B f v / S u d 0 z D c I W 1 f u w Y K o f U G J P 8 D 8 8 7 n v g u V X h v n a M B c m K t W Z 4 i v 6 M I R h j P F 7 v 0 E V 7 z C P O p / 7 G T 9 w g l N t U K t q c 9 r 7 T q r W V W i e 4 b f Q a r f n v J x q < / l a t e x i t > (a.p.) : p 0 = E a p FIG. 2. A schematic picture of the dispersion relations of q.p., p0 = E + p (red solid line), q.h., p0 = E − p (blue dashed line), and a.p., p0 = E a p (purple dotted line).
U (p)'s, V (p)'s, and W (p)'s in Eq. ( 7) are The dispersion relations (10) and the factors (11) clearly show that only the positive-energy component of light quarks couples to the Kondo condensate.It should be noted that U ± (p) and W ± (p) satisfy U + (p) + U − (p) = 1 and W + (p) + W − (p) = 1, respectively.A schematic picture of the dispersion relations of q.p. (red solid line), q.h.(blue dashed line), and a.p. (purple dotted line) is shown in Fig. 2.This figure shows that q.p. always lives above the Fermi level, while q.h. and a.p. are below the Fermi level.The energy gap between q.p. and q.h. at the identical momentum p satisfies where the minimum value is given at |p| = µ.In other words, the threshold energy of pair creation or pair annihilation of q.p. + q.h. at rest frame is found to be δE min.= 8|∆| 2 as indicated in Fig. 2. From the Lagrangian (3) with the ansatz (4) inserted, the thermodynamic potential in the absence of the gauge field is given by with the three-dimensional volume and the inverse temperature, V and β = 1/T , respectively.The cutoff parameter Λ is included to regularize the integral, and N c is the number of colors which is taken to be N c = 3.By taking a derivative of the thermodynamic potential (13) with respect to ∆, the gap equation that the Kondo condensate satisfies is derived as with the Fermi-Dirac distribution function and hence an approximate solution of Eq. ( 14) at T = 0 can be found as [9,13] with the density of states of a massless quark at the Fermi level, ρ(µ) = µ 2 2π 2 , under assumptions of ∆ µ, Λ and N c gρ(µ) 1.The solution (15) shows that the Kondo condensate ∆ significantly appears in cold dense quark matter.The integration in Eq. ( 14) is mostly governed by modes near the Fermi sphere for the noninteracting case |p| ∼ µ.This fact allows us to estimate the temperature dependence of ∆ in an easy way.Namely, the factor by distribution functions f F (E + p ) − f F (E − p ) at the Fermi sphere turns into Here, if T √ 2∆, then thermal fluctuations are rather suppressed, leading to which is the same value as that at T = 0.In other words, because of the gapped nature of q.p. and q.h., at lower temperature, the magnitude of ∆ does not change significantly from that at zero temperature.Numerically such a tendency was found in Ref. [14].We employ the gap equation ( 14) to determine the size of the Kondo condensate in the following analysis, since the magnetic field is treated perturbatively based on the linear response theory.

III. ANALYTIC EVALUATION
In this section, we give analytic evaluation of the HQSP induced by the Kondo effect under a magnetic field with vertex corrections required by the U (1) EM gauge symmetry.

A. General properties
The emergence of the HQSP is described by a thermodynamic expectation value of a spin polarization of the heavy quarks.For this purpose, we define with S i h = σ i 2 being a spin operator of heavy quarks.Introducing six-component spinors Φ and Φ containing the light-and heavy-quark fields, Equation ( 18) can be expressed in terms of Φ and Φ as where S i H is the 6×6 matrix spin operator of heavy quark One of the useful ways to calculate the expectation value (20) is the analytic continuation from the imaginary-time formalism [36].Namely, first we calculate the expectation value of Si H (iω n , q) β with ωn = 2nπT the Matsubara frequency for bosons (n is an integer), and next, we evaluate Eq. ( 20) by the following relation: In this equation, η is an infinitesimal positive number.
With the help of linear response theory, Si H (iω n , q) β at the leading order of the gauge field can be given as in which we have defined Ãµ (iω n , q) as the Fourier transformation of A µ (t, x) within the imaginary-time formalism.Here, ω m = (2m + 1)πT is the Matsubara frequency for fermions (m is an integer), and iω m ≡ iω m + iω n , p ± ≡ p± q 2 .3Note that Γ µ in Eq. ( 23) is the vertex function responsible for the coupling between the fermions and the gauge field.Regarding the spin polarization (23), the response function of the HQSP to a magnetic field ΠH (iω n , q) is defined by where Bi (iω n , q) is the magnetic field generated by the gauge field Ãµ (iω n , q).

B. Gauge invariance and vertex corrections
As pointed out in our previous work [25], naive adoption of the bare vertex read by the Lagrangian in Eq. ( 1) or Eq. ( 3) leads to the violation of U (1) EM gauge symmetry.One useful way to cure this problem is to correct the vertex Γ µ in an ad hoc manner such that it recovers the U (1) EM invariance.Here, we show the detailed procedure of this treatment in the real time.
The U (1) EM gauge invariance of the Lagrangian (1) yields the Ward-Takahashi identity for the vertex (Γ µ ) between the fermions and the gauge field as [37] with p ± 0 = p 0 ± q0 2 , where G−1 0 (p 0 , p) has been defined in Eq. ( 5), and Q is the charge matrix Therefore, by denoting the vertex by the identity ( 25) is explicitly given by Equations ( 28) and (31) show that we can safely employ the bare vertex as for the diagonal components of Eq. ( 27).On the other hand, the off-diagonal ones no longer vanish and have to be corrected to satisfy Eqs. ( 29) and (30).
In a later analysis, we study the HQSP for a small external momentum q p.This restriction is reasonable since the loop integral in Eq. ( 23) is dominated by modes around |p − | ∼ |p + | ∼ µ or those having sufficiently large density of states; the modes close to p ∼ 0 do not contribute to the HQSP significantly.Hence, we expand the right-hand sides (rhs's) in Eqs. ( 29) and (30) up to O(q 1 ). 4 In this approximation, the identities ( 29) and ( 30) are reduced to 4 In order to treat this expansion symmetrically we have defined the spatial part of loop momenta as in Eq. ( 23): and respectively.When we take a q µ → 0 limit in Eqs.(33) and (34), the second terms of the rhs's vanish while the first terms survive.The nonvanishing behavior of the first term implies that radiative contributions by a massless mode couple to the vertices.Namely, the first terms in Eqs. ( 33) and ( 34) are responsible for the Nambu-Goldstone (NG) mode contributions [37].In fact, such contributions arise only when the Kondo condensate is electrically charged (e q = e Q ) where the U (1) EM symmetry is spontaneously broken. 5On the other hand, the second terms in Eqs. ( 33) and ( 34) are important even when the condensate is electrically neutral (e q = e Q ) since they are proportional to e q +e Q .Namely, regardless of the spontaneous breakdown of the U (1) EM symmetry, the vertex corrections are necessary.Those contributions appear due to the nontrivial momentum dependence of the hedgehog ansatz in Eq. ( 4).According to discussions in Appendix B, general solutions of Eqs. ( 33) and (34) relevant to the magnetic response are given by respectively.In these equations, Γ− A ψΨv and Γ+ are arbitrary scalar functions of p and q which cannot be fixed (for more detail, see Appendix B).They appear because the identities (33) and (34) include the combinations of q µ Γ µ A ψΨv and . In other words, Γ− A ψΨv and Γ+ could be regarded as "integration constants".In the following analysis, we choose those values to be zero: This choice preserves the Hermiticity of the HQSP by the vertex corrections.Here, we note that the vertices proportional to q j have not been included in Eqs.(35) and (36) since such contributions vanish due to the transversality of the vertices as explained in Appendix A.
For the same reason, the NG mode contributions disappear because they are longitudinal.
In this subsection, we have shown that the U (1) EM invariance requires us to employ the corrected vertices (35) and (36) in addition to the bare ones (32).Based on them, in Sec.III C, we proceed with evaluation of the HQSP (23).
C. Evaluation of Eq. ( 23) Here, we proceed with analytical evaluation of the HQSP in Eq. ( 23) with the corrected vertices obtained in Sec III B.
Since the magnetic field enters only the spatial components of A µ , we take A 0 = 0. Thus using Eq. ( 27), Eq. ( 23) is given by the following two parts: Here Si 0H (iω n , q) β stands for the HQSP obtained through the bare vertex in Eq. (32) as and Si δH (iω n , q) β through the vertex corrections in Eqs. ( 35) and (36) with Eq. ( 37) inserted as Correspondingly, the HQSP response function to the magnetic field given in Eq. ( 24) is separated as ΠH (iω n , q) = Π0H (iω n , q) + ΠδH (iω n , q) , (41) with each response function defined by Si 0H (iω n , q) β = e Bi (iω n , q) Π0H (iω n , q) , (42) Si δH (iω n , q) β = e Bi (iω n , q) ΠδH (iω n , q) .( In Eqs.(39) and (40), the contribution proportional does not exist.This means that the magnetic field does not couple to the heavy quark directly; the HQSP in the absence of Kondo condensate vanishes.It is also easily understood by the lack of magnetic coupling between the heavy quark and the gauge field in Eq. ( 1).
In the following calculation, we treat the bare-vertex part (39) in detail as a demonstration.From the Green's function in Eq. ( 7) together with the bare vertex in Eq. ( 32), the calculation of Eq. ( 39) is further proceeded as with In getting Eq.( 44), we have made use of the Matsubara summation formula The constituent (45) includes information on the quantum and thermal fluctuations of each process upon the Kondo effect, which is not specific to the response to a magnetic field.Namely, the constituent (45) always appears when we evaluate such fluctuations at one loop regardless of the detail of interactions.On the other hand, the kinetic factor tr[σ i Λ Hp (p + )γ j Λ pH (p − )] in Eq. ( 44) reflects information on the couplings among the spins of the light quark, heavy quark, and gauge field, contributing to the HQSP.By making use of a trace formula FIG. 3. The Feynman diagrams contributing to the HQSP.Blobs represent the corrected vertices required by U (1)EM gauge invariance.The diagrams (i) and (ii) correspond to the intraband processes while (iii) and (iv) the interband ones.
Eq. ( 44) turns into In obtaining the second equality, we have used Bi (iω n , q) = i ijk q j Ãk (iω n , q) and defined êq via e q = êq e.Therefore, comparing Eqs. ( 42) and ( 48), the response function from the bare-vertex part Π0H (iω n , q) is evaluated as with In a similar way, the response function from the corrected-vertex part ΠδH (iω n , q) in Eq. (43) reads with where êQ is defined via e Q = êQ e.In Eq. ( 52) we have defined the constituent that accounts for only quantum and thermal fluctuations stemming from the vertex corrections as an analogue of Eq. ( 45).As explained in Sec.III B, the vertex corrections are necessary although the NG mode does not couple to a magnetic field, because of the momentum dependence of the Kondo condensate.Indeed, Eq. ( 52) accounts for such corrections to the HQSP.The Feynman diagrams for Π++ H , Π−− H , Π+− H , and Π−+ H are depicted in Fig. 3.These contributions correspond to (i) q.p. → q.p. scattering, (ii) q.h.→ q.h.scattering, (iii) q.p. + q.h.pair annihilation, and (iv) q.p. + q.h.pair creation, respectively.The diagrams (i) and (ii) are often referred to as the intraband processes, while the remaining (iii) and (iv) are as the interband processes. 6In this figure blobs represent the corrected vertices.The HQSP Si H (q 0 , q) β in the real time can be obtained via the analytic continuation in Eq. ( 22).In the same manner, the response function ΠH (q 0 , q) in the real time can be evaluated.In our present paper, we investigate the HQSP response functions with vertex corrections ΠH (q 0 , q) = Π0H (q 0 , q) + ΠδH (q 0 , q) , ( and without them Π0H (q 0 , q), to examine the importance of gauge invariance clearly.In what follows, we take a phase of ∆ such that ∆ is always real without loss of generality.

IV. THE HQSP IN THE DYNAMICAL AND STATIC LIMITS
In our present paper, we investigate the HQSP response function for vanishing spatial momentum Π0H (q 0 , 0) and ΠH (q 0 , 0) (timelike) and for vanishing frequency Π0H (0, q) and ΠH (0, q) (spacelike) toward understanding of the HQSP in the two distinct momentum regions in the clearest way.Physically, the former (latter) describes the HQSP whose time dependence is faster (slower) than the equilibration of the spatial part of the system.In particular, it is useful to examine the response function in the dynamical limit, and in the static limit in detail to see differences in the two momentum regimes [38].Before moving on to numerical computations, in this section, we analytically study the dynamical and static response functions in Eqs. ( 55) and (56).
A. The bare-vertex parts Π0H Here, we evaluate analytically the HQSP response functions Π0H (q 0 , q) from the bare-vertex parts.The q 0 (or iω n in the imaginary time) dependence of the response functions is rather trivial as can be seen from Eq. ( 45), and hence we expand Π0H (q 0 , q) with respect to a small momentum q.
From Eq. ( 45), I ζζ 0H (p + ; p − ) with small q for the intraband processes described by ζ = ζ can be evaluated as while for the interband ones described by ζ = ζ as Besides, the common kinetic factor p− • q − p+ • q is expanded as Hence, the response functions with a small momentum q are of the forms for the intraband processes, while for the interband processes.Equation (60) shows that the HQSP response functions in the real time for the intraband processes lead to distinct results in the dynamical and static limits, and Equation (62) indicates that Πζζ 0H in the dynamical limit is always zero.In the static limit, at zero temperature ∂f F (E ζ p )/∂E ζ p in Eq. ( 63) becomes zero since the density of states of q.p. or q.h.vanishes at the Fermi level, and hence At finite temperature, the factor ∂f F (E ζ p )/∂E ζ p begins to collect contributions around the Fermi level by thermal fluctuations.In the present Kondo system, due to the flatness of the dispersion relation of q.p. or q.h. as shown in Fig. 2, the thermal fluctuations can pick up modes slightly far from the Fermi sphere but having large density of states.As a result, Πζζ 0H in the static limit at finite temperature becomes nonzero.
On the other hand, from Eq. ( 61), we can find that the HQSP response functions for the interband processes in the two limits yield the identical result, lim q→0 Πζζ 0H (0, q) = lim q0→0 Πζζ 0H (q 0 , 0) condensates equally contribute, the resultant HQSP response function is given by the averaged value of Figs. 4  and 5, yielding a finite negative response.Next, we show numerical results of q 0 dependence of the response function for vanishing q at µ = 0.5 GeV. Figure 6 represents the results of Π0H (q 0 , 0) (dashed purple line) and ΠH (q 0 , 0) (solid blue line) for êq = + 2 3 at T = 0 and T = 0.03 GeV.Similarly, in Fig. 7, we show Π0H (q 0 , 0) (dashed purple line) and ΠH (q 0 , 0) (solid red line) for êq = − 1 3 .Note that we plot the result for only q 0 > 0, since the response function is symmetric with respect to an inversion of q 0 → −q 0 as ΠH (q 0 , 0) = ΠH (−q 0 , 0), due to time-reversal symmetry of the Kondo phase.
In what follows, we discuss the divergent behaviors near q 0 ∼ 0.2 GeV in Figs. 6 and 7, where the difference between Π0 (q 0 , 0) and ΠH (q 0 , 0) becomes more significant.The reason is as follows.The threshold energy for the pair annihilation and pair creation of q.p. and q.h.[the diagrams (iii) and (iv) in Fig. 3] is given by δE min = √ 8∆ 2 , as indicated in Fig. 2. Thus, when q 0 becomes larger than δE min = √ 8∆ 2 , the imaginary part corresponding to the above physical processes appears.The value of threshold energy is δE min ≈ 0.239 GeV for T = 0 and δE min ≈ 0.210 GeV for T = 0.03 GeV, at µ = 0.5 GeV.Above this threshold, we encounter a di-vergence that cannot be regulated by the UV cutoff Λ, when we include the vertex corrections as well.This divergence stems from the interband processes of correctedvertex parts of the form Πζζ δH (q 0 , 0) [see Eq. (C3) for the explicit expression].Namely, the divergence arises from the factor 1/(q 0 + iη − E ζ p + E ζ p ) 2 and cannot be removed by, e.g., making use of the principal value integral.For this reason, in Figs. 6 and 7, we have shown the results where q 0 is smaller than the threshold.
In more realistic situations, the problematic factor 1/(q 0 + iη − 2 due to a finite relaxation time τ R .In this case, the divergence will be smeared.In field-theoretical treatments, the relaxation time τ R can be evaluated by a self-energy of the Green's function of fermions beyond the perturbative calculation.Namely, for a feasible treatment in the higher frequency regime, we need to employ a nonperturbative method to determine the self-energy, such as the Dyson-Schwinger equations coupling with the Ward-Takahashi identity (25).Although the behavior above the threshold is problematic in our present treatment, we can see a significant enhancement of the HQSP as the frequency q 0 approaches the threshold from below, which is a universal behavior of response functions.

B. The HQSP response function in the spacelike regime
In this subsection, we show the µ dependence of the response function in the spacelike regime.
First, we examine the response function in the static limit defined in Eq. (55).In this limit, the intraband processes in addition to the interband ones, namely, all of the diagrams in Fig. 3 contribute.Figure 8 represents the results of Πsta 0H (dashed purple line) and Πsta H (solid blue line) for êq = + 2  3 at T = 0, T = 0.01 GeV and T = 0.03 GeV.Similarly, in Fig. 9 we show Πsta 0H (dashed purple line) and Πsta H (solid red line) for êq = − 1 3 .As in the dynamical limit, the vertex corrections enhance the HQSP response function for êq = êQ , while they suppress the HQSP for êq = êQ .At finite temperature, the HQSP in the static limit is suppressed compared to that in the dynamical limit for any electric charges.This suppression shows that the intraband processes driven by the mechanism explained in Sec.IV are considerably large.It is worth noting that such intraband contributions at finite temperature were also found in the CSE with the Kondo effect [25].In addition, we can see that the vertex corrections becomes relatively insignificant at T = 0.03 GeV.
Here, we summarize differences between the HQSP with vertex corrections in the dynamical limit (Figs. 4  and 5) and that in the static limit (Figs. 8 and 9).At zero temperature, the Πdyn

H and Πsta
H coincide, while at finite temperature they differ, which is consistent with the properties found analytically in Sec.IV.Besides, nu- merically, we have found that the magnitude of Πdyn H is always larger than that of Πsta H for êQ = êq = + 2 3 .On the other hand, for êQ = + 2  3 and êq = − 1 3 , the difference of magnitudes depends on temperature.
Next, we show numerical results of |q| dependence of the response function for vanishing q 0 at µ = 0.5 GeV.Because of the Gauss's law for the magnetic field, the momentum q is transverse to the field: q i Bi (0, q) = 0. Figure 10 represents the results of Π0H (0, q) (dashed purple line) and ΠH (0, q) (solid blue line) for êq = + 2  3 at T = 0 and T = 0.03 GeV.Similarly, in Fig. 11 we show Π0H (0, q) (dashed purple line) and ΠH (0, q) (solid red line) for êq = − 1 3 .It should be noted that we have plotted the results for the momentum up to q = 0.3 GeV, while we assumed that q is sufficiently small compared to µ.
Unlike the frequency dependence of the HQSP response function studied in Sec.V A, Figs. 10 and 11 show that the magnitude of ΠH (0, q) does not change significantly.This is because there is no notable kinetic effects such as the existence of thresholds in the spacelike region.In particular, for the small |q| regime such a stable behavior is understood well.Namely, since the loop integrals in Eqs. ( 50) and ( 52) are dominated by modes at |p + | ∼ |p − | ∼ µ or those having sufficiently large density of states, the small q does not change the HQSP signifi-cantly.

VI. THE HQSP FROM THE ZEEMAN INTERACTION
Up to this point we have investigated the HQSP induced by the Kondo effect under a magnetic field which arises at O(1/m 0 Q ) within the HQET.When we go beyond this order, the ordinary Zeeman interaction (ZI) at O(1/m 1 Q ) is expected to become another source of the HQSP.Hence, in this section we discuss the HQSP induced by the ZI in the absence of the Kondo effect and compare its magnitude with that driven through the Kondo effect of O(1/m 0 Q ).In the following analysis, we employ the grand canonical picture such that the chemical potential of heavy quarks measured within the nonrelativistic framework is always zero as done for the Kondo effect in this paper.
In the Lagrangian (1), we have described the heavy quark within the leading order of the HQET, where only terms of O(1/m 0 Q ) have been taken into account.When we include contributions of O(1/m 1 Q ) as well, the Lagrangian for the heavy quark can be given by Here, we have left only the kinetic and ZI terms with the spin operator for the heavy quark S h = σ 2 .As done in Sec.III, from Eq. (68), the HQSP by the ZI under the weak magnetic field within the linear response theory is evaluated as where is the Green's function for the heavy quark in the nonrelativistic framework with E NR p = |p| 2 /(2m Q ), and p = p + q.Performing the Matsubara summation with Eq. ( 46), the HQSP (69) with a small q reads SZI,i H (iω n , q) ≈ N c e Q Bi (iω n , q) Thus, defining the response function ΠZI H (iω n , q) via SZI,i H (iω n , q) = e Bi (iω n , q) ΠZI H (iω n , q) , (72) ΠZI H (iω n , q) in the dynamical and static limits in the real time are evaluated as lim q0→0 ΠZI H (q 0 , 0) = 0 , Equation (73) shows that ΠZI H in the dynamical limit vanishes, because interband processes for the HQSP driven by the ZI are absent.In other words, the emergence of HQSP in the dynamical limit can be regarded as a peculiar phenomenon induced by the Kondo effect under magnetic field.From Eq. (74) we can see that ΠZI H in the static limit is always positive for finite T .The positive ΠZI H is understood as follows: from the ZI term in Eq. (68), the corresponding ZI Hamiltonian is always negative when the heavy-quark spin and the magnetic field are parallel for e Q = + 2 3 .Thus, the positive spin polarization is thermodynamically favored more than the negative one, resulting in the positive HQSP.Besides, ΠZI H at T = 0 is zero, since in the three-dimensional space ΠZI H is of O(B 3/2 ) and can be neglected within the linear response theory.
In Fig. 12, we compare the T dependence of the HQSP response function induced by the Kondo effect and that by the ZI in the static limit.The solid blue line corresponds to the result by the Kondo effect ( ΠH ) with êQ = êq = + 2 3 , while the solid red one with êQ = + 2 3 , êq = − 1 3 , at µ = 0.5 GeV (recall that this µ is the chemical potential for light quarks).The dotted pink line shows the result by the ZI ( ΠZI H ) for the c quark with m Q = 1.27GeV. Figure 12 shows that the HQSP is clearly dominated by the contribution from the Kondo effect at lower temperature particularly for êQ = êq = + 2 3 even in the static limit.

VII. CONCLUSIONS
In this paper, we have proposed a new mechanism of the HQSP in quark matter induced by the Kondo effect under a magnetic field.By employing the NJL type model, we have indeed shown that the HQSP is driven through the Kondo condensate from light quarks coupling with the magnetic field, although a magnetic coupling of the heavy quarks themselves is absent in the heavy-quark limit.In particular, we have demonstrated the emergence of HQSP in the distinct momentum regimes: the timelike and spacelike momentum regions.Physically, the former (latter) describes the HQSP whose time dependence is faster (slower) than the equilibration of the spatial part of the system.The effects of vertex corrections required by the U (1) EM gauge symmetry have been also examined.
Our analysis shows that the response function of the HQSP is significantly driven in both the momentum regimes.Also, we have found that the vertex corrections enhance the resultant HQSP compared to that with the bare vertices, when the Kondo condensate is electrically neutral.In addition, the timelike HQSP is significantly enhanced as the frequency of the magnetic field approaches the threshold of pair creation (annihilation) of the quasiparticle and quasihole induced by the Kondo effect, whereas the spacelike one does not vary largely with the momentum.
We have also discussed the HQSP induced by the Zeeman interaction of heavy quarks, as corrections from violation of the heavy-quark limit.As a result, we have found that the HQSP induced by the Zeeman interaction in the dynamical limit (zero frequency and momentum limits from the timelike regime) always vanishes and that in the static limit (similar limit from the spacelike regime) at lower temperature is largely suppressed.Therefore, emergence of the HQSP particularly in such regimes can be a useful signal of the Kondo effect.
Experimentally, the peripheral and low-energy heavyion collisions (HICs) are expected to become a testing ground to investigate the HQSP induced by the Kondo effect.In fact, in such HICs heavy quarks as impurities are produced by hard processes mediated by gluons, together with a magnetic field and a sufficient quark chemical potential.In this case, the HQSP is converted into the spin polarization of heavy hadrons which can be observables [23].
The HQSP together with the emergence of the Kondo condensate can be examined in detail in future lattice simulations of QCD (or QCD-like theories).The HQSP is measured by computing the operator Ψ † (t, x) σ 2 Ψ(t, x) , where Ψ(t, x) and σ are the heavy-quark field and the Pauli matrix parallel to a magnetic field B, respectively.For the lattice simulations, one has to introduce, in addition to the usual QCD, the three additional backgrounds: (i) a nonzero chemical potential for light quarks, (ii) a magnetic field, and (iii) heavy impurities.For (i), although Monte Carlo simulations for systems with a nonzero quark chemical potential are usually useless due to the sign problem, one may utilize other sign-problemfree systems, such as two-color QCD, isospin chemical potential µ I , and chiral chemical potential µ 5 . 8Such simulations have been devoted to elucidating the properties of dense QCD in a background magnetic field [55][56][57][58], and model studies to intuitively understand the simulations have been also done [59][60][61][62][63][64][65].For example, our present analysis can be easily applied to the two-color system by changing the number of colors from N c = 3 to N c = 2. Namely, our proposal in this paper is testable in two-color QCD simulations without major changes.In particular, the static three-dimensional momentum dependence of the HQSP response function within the linear response regime can be directly measured by the lattice simulations.For (iii), we note that inclusion of heavy quarks as impurities does not spoil the sign-problem-free advantage unless we introduce a "chemical potential" for the heavy quarks as done in this paper.We expect that a better understanding of the role of heavy impurities and their properties under external fields in quark matter will be promoted by such simulations in the future.
Keeping in mind the above remarks, after a lengthy but straightforward calculation, Πζζ δH (iω n , q) for the in-traband and interband processes with a small momentum q are reduced to Π ζζ δH (q) ≈ −∆ (ê q + êQ )N c 2 and Π ζζ δH (q) ≈ ∆ (ê q + êQ )N c 2 respectively.It should be noted that unfamiliar contributions including 1/(q 0 − E ζ p + E ζ p ) 2 appear in Eq. (C3).They cause divergences for higher q 0 which cannot be removed by the UV cutoff Λ. Taking low momentum limits properly in Eq. (C2), the response functions from the corrected-vertex parts for the intraband processes in the dynamical and static limits read respectively.The vanishing result in the dynamical limit and the contributions proportional to ∂f F (E ζ p )/∂E ζ p in the static limit are the same as those from the bare-vertex parts.Similarly, from Eq. (C3) the response function for the interband processes in the dynamical and static limits are evaluated as lim q→0 Πζζ 0H (0, q) = lim q0→0 Πζζ 0H (q 0 , 0) Namely, similarly to the bare-vertex parts, the interband processes yield the identical results in the two limits.
r X O Z N 7 G A P W R T 5 u g b 6 O I s 8 S b K U G P 2 T F B l / W A J f Q l r 6 A A Y J h 7 c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " N z Fi 7 f W s y G e k + i e / C H E K / 8 X Z N C o = " > A A A C h X i c h V F N S x t B G H 6 y t W r j 1 z a n g p f Q a I m I y 7 t e D I H S F i l 4 N L F R w e i y u 0 7 M k v 3 q 7 i T U L v k D v f X U Q 0 8 K U k p / g N c W L / 6 B H v w J 4 l G h l x 7 6 Z h M o b V D f Y W a e e W a e d 5 6 Z 1 w p d J 5 Z E F x n l w c j D 0 b H x R 9 m J y a n p G f X x 5 G Y c t C N b 1 O z AD a J t y 4 y F 6 / i i J h 3 p i u 0 w E q Z n u W L L a q 3 2 9 r c 6 I o q d w H 8 j D 0 O x 6 5 k H v t N w b F M y Z a i L x a Q e e f m 3 W l P r L p R D g 5 6 / N p e e W a e d 5 6 Z 1 w p d J 5 Z E F x n l w c j D 0 b H x R 9 m J y a n p G f X x 5 G Y c t C N b 1 O z A D a J t y 4 y F 6 / i i J h 3 p i u 0 w E q Z n u W L L a q 3 2 9 r c 6 I o q d w H 8 j D 0 O x 6 5 k H v t N w b F M y Z a i L x a Q e e f m 3 W l P r L p R D g 5 6 / N p y s p N 9 g T o y r K 1 y l u A M g 9 k d H r d 5 t Z 6 y H q 8 b N a N E b f E p D v e Q l T m M 0 i 8 6 o x u 6 o h / 0 m + 6 e r R U n N R p e 9 n g 2 m 1 o R 6 H 3 7 Q 8 u 3 / 1 W 5 P E t U 7 l U v e p b Y Q i H x a r P 3 I G E a t 7 C a + t q 3 g 5 v l 4 t J o / J a O 6 Q / 7 P 6 J r u u Q b e L W / 1 s m i W D p E h j 9 A e / z c T 8 H K u K q R q i 1 S f v p j + h W d G M Y b j P F 7 T 2 A a 8 1 h A i c / 9 j p 8 4 x 4 U y o B S U K W W m m a q 0 p J p B P A j l 0 z / G R Z l R < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " K k c Q 7 1 y B N h o T t U X P o S 2 E O j o 9 h 2 s = " > A A A C k H i c S y r I y S w u M T C 4 y c j p x 6 z t p T K U p T D S P 4 E c g Q y i y N u p a + x g H z Y 0 1 G B C w I L P 2 I A C j 9 s 2 Z B A c 5 n b R Z M 5 l p I f 7 A s d I s L b G W Y I z F G a r P F Z 4 t R 2 p x 6 z t p T K U p T D S P 4 E c g Q y i y N u p a + x g H z Y 0 1 G B C w I L P 2 I A C j 9 s 2 Z B A c 5 n b R Z M 5 l p I f 7 A s d I s L b G W Y I z F G a r P F Z 4 t R 2 p x 6 z t p T K U p T D S P 4 E c g Q y i y N u p a + x g H z Y 0 1 G B C w I L P 2 I A C j 9 s 2 Z B A c 5 n b R Z M 5 l p I f 7 A s d I s L b G W Y I z F G a r P F Z 4 t R 2 d b a m M / i u 6 a I d r 0 D / g D I W S V g I j k I 7 J w o x / g w k 8 I L h X c u P B O T 0 N I J M k t q u r U q X t u n a p y p O 8 l i u i 8 S + v u 6 b 1 3 v + 9 B 6 W H / o 8 c D 5 S d P l 5 O o G b u i 7 k Z + F K 8 6 d i J 8 L x R 1 5 S l f r M p Y 2 I H j i x V n Z 7 a 9 v 9 I S c e J F 4 Z L a l W I 9 s L d C b 9 N z b c W
FIG.12.The HQSP response function induced by the Kondo effect and that by the ZI as functions of T .For details see the text.